Geometry of Matrices: In Memory of Professor L K Hua

GEOMETRY

Of: MATRICES

This page is intentionally left blank

GEOMETRY OF MATRICES In Memory of Professor L K Hua (1910 -1985)

Zhe-Xian Wan Chinese Academy of Sciences, China Lund University, Sweden

{World Scientific Singapore'New Jersey • L Singapore* NewJersey•London •Hong Kong

Published by World Scientific Publishing Co Pte Ltd P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Wan, Che-hsien. Geometry of matrices / Zhe-xian Wan. p. cm. "In memory of Professor L. K. Hua (1910-1985)." Includes bibliographical references and index. ISBN 9810226381 1. Matrices. I. Title. QA188.W36 1996 516.3'5--dc20

96-2179 CIP

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Preface The present monograph is a state of the art survey of the geometry of matri­ ces. Professor L. K. Hua initiated the work in this area in the middle forties. In this geometry, the points of the space are a certain kind of matrices of a given size, and the four kinds of matrices studied by Hua are rectangular matrices, symmetric matrices, skew-symmetric matrices and hermitian ma­ trices. To each such space there is associated a group of motions, and the aim of the study is then to characterize the group of motions in the space by as few geometric invariants as possible. At first, Professor Hua, relating to his study of the theory of functions of several complex variables, began studying the geometry of matrices of various types over the complex field. Later, he extended his results to the case when the basic field is not necessar­ ily commutative, discovered that the invariant "adjacency" alone is sufficient to characterize the group of motions of the space, and applied his results to some problems in algebra and geometry. Professor Hua's pioneer work in the area has been followed by many mathematicians, and more general results have been obtained. I think it is now time to summarize all results obtained so far, and this has been my motivation for the present work. In order to be as self-contained as possible the book covers some material of linear algebra over division rings in Chapter 1, which is necessary for later chapters. This chapter can also be read independently as an introduction to linear algebra over division rings. The fundamental theorems of the affine geometry and of the projective geometry over any division ring constitute the main contents of Chapter 2. In particular, Hua's beautiful theorem on semi-automorphisms of a division ring and its application to the fundamental theorem of the one-dimensional projective geometry over a division ring are v

PREFACE

VI

included. Following these chapters, the geometry of rectangular matrices over any division ring, alternate matrices over any field, symmetric matrices over any field, and the geometry of hermitian matrices over any division ring which possesses an involution are discussed in detail in Chapters 3, 4, 5, and 6, respectively. Applications to problems in algebra, geometry, and graph theory are included throughout. Finally, the author is indebted to Yangxian Wang and Mulan Liu for their helpful comments on the first draft of the book, to Rongquan Feng, Lei Hu, Xinwen Wu, and Zhanfei Zhou for their laborious typewriting, and to Lena Mansson for her beautiful improvement of the camera-ready copy. Zhe-xian Wan

Contents Preface

v

1

Linear Algebra over Division Rings 1.1 Vector Spaces over Division Rings 1.2 Matrices over Division Rings 1.3 Matrix Representations of Subspaces 1.4 Systems of Linear Equations 1.5 Hermitian, Symmetric, and Alternate Matrices 1.6 Comments

1 1 11 27 29 35 44

2

Affine G e o m e t r y and Projective G e o m e t r y 2.1 Affine Spaces and Affine Groups

45 45

2.2 2.3 2.4 2.5 2.6

54 66 76 80 87

3

Fundamental Theorem of the Affine Geometry Projective Spaces and Projective Groups Fundamental Theorem of the Projective Geometry One-dimensional Projective Geometry Comments

G e o m e t r y of Rectangular Matrices 3.1 The Space of Rectangular Matrices 3.2 Maximal Sets of Rank 1 3.3 Maximal Sets of Rank 2 3.4 Proof of the Fundamental Theorem 3.5 Application to Algebra 3.6 Application to Geometry 3.7 Application to Geometry (Continued) vn

89 89 93 97 106 118 123 139

viii

CONTENTS 3.8 3.9

Application to Graph Theory Comments

153 155

4

G e o m e t r y of Alternate Matrices 4.1 The Space of Alternate Matrices 4.2 Maximal Sets 4.3 Proof of the Fundamental Theorem 4.4 Application to Geometry 4.5 Application to Geometry (Continued) 4.6 Application to Graph Theory 4.7 Comments

157 157 159 168 177 199 211 215

5

G e o m e t r y of Symmetric M a t r i c e s 5.1 The Space of Symmetric Matrices 5.2 Maximal Sets of Rank 1 5.3 Maximal Sets of Rank 2 (Characteristic Not Two) 5.4 Proof of the Fundamental Theorem (I) 5.5 Maximal Sets of Rank 2 (Characteristic Two) 5.6 Proof of the Fundamental Theorem (II) 5.7 Proof of the Fundamental Theorem (III) 5.8 Application to Algebra 5.9 Application to Geometry 5.10 Application to Graph Theory 5.11 Comments

217 217 222 224 231 244 252 264 281 285 296 303

6

G e o m e t r y of Hermitian Matrices 6.1 The Space of Hermitian Matrices 6.2 Maximal Sets of Rank 1 6.3 Maximal Sets of Rank 2 6.4 Proof of the Fundamental Theorem (the Case n > 3) . . . . 6.5 Maximal Sets of Rank 2 (the Case n = 2) 6.6 Proof of the Fundamental Theorem (the Case n = 2) . . . . 6.7 Application to Algebra 6.8 Application to Geometry 6.9 Application to Graph Theory

305 305 308 311 323 341 348 355 356 363

CONTENTS 6.10 Comments

ix 365

Bibliography

367

Index

371

Chapter 1 Linear Algebra over Division Rings 1.1

Vector Spaces over Division Rings

Let D be any division ring and n a positive integer. We use D{n) = {On, a?2, • • •, i n ) I Xi G D, i = 1, 2, • • •, n} to denote the n-dimensional row vector space (or left vector space) over D formed by the set of all n-tuples (or n-dimensional row vectors) (xu

x 2 , • • •, x n ),

Xi£ D, i = 1, 2, • • •, n,

over D with addition and scalar multiplication defined by (xu x 2 , • • •, xn) + (2/1,2/2, * • •, 2/n) = («i + 2/1, x2 + 2/2, • • • 9xn + yn) and X\Xiy

X 2 , * " * , Xfij

— y*LXi)

aJX 2 ,

,

d/Xjij^

respectively, where x, xi, ar2, •••, xn, ?/i, ?/2, •••, Vn € D. For any row vectors u, u, iu G i ^ and re, y G D we have the following manipulation rules U + V = V + U,

(u + v) + w = u + (v + w), x(?i + v) = xu + xv, (x + 2/)^ = a;w + yu, (xy)u = x(yu), lu = u, 1

2

Chapter 1. Linear Algebra over Division Rings

where 1 is the identity element of D. Moreover, let dci = (l,0,0,-( l , 0 , 0 , - - -•,0), , 0 ) , e2 = (0,1,0,-• (0,l,(),•••,0), ( 0 , 0 , - -•,o,i), -,0,l), •,o), •••••,•, ee nn = (0,0,-then ei, e 2 , •••, e n form a basis of D^n\ i.e., any vector (#i, £ 2 , •••, # n ) of D^ can be written as a linear combination of ei, e 2 , • • •, e n with coefficients in D: (xi, xZ22,, •' • • •, •, xnn)) = = xiei xnen, xxex + + zx22ee22 H + • •~r\~ ^n^nt 0*1, and the expression is unique. Let v = (ai, a 2 , • • •, a n ) be an n-dimensional row vector of D^n\ then a t (1 < i < n) is called the i-th component of v. If all the components of v are 0, i.e., v=(0, 0, • • •, 0), then v is called the zero (row) vector and denoted simply by 0, i.e., (0, 0, 0)==0.0. (o, o, • •■•,•, 0) Clearly, for any row vector v € D^

we have

V. v+ 0= == 0 + v === v.

Let v = (ai, a 2 , • • •, an) G D^

and define

-v - v = (-ai, (—ai, --aa 2 ,, •••, • •» • —a - f l nn)),, then vw + ((-v) 0. --v)-- == 0. Definition 1.1: Let t>i, v 2 , • • •, vT be r row vectors in D^1'. They are said to be linearly dependent (over D) if there are r elements ai, a 2 , • • •, ar of Z), which are not all equal to 0, such that a\Vi + a2v22 + ai^i + •-••h a+rvaTrvr= = 0; 0; otherwise, they are said to be linearly independent

□

It follows immediately from Definition 1.1 that any finite number of row vectors in , among which the zero vector 0 appears, are linearly depen­ dent and that the vectors of any nonempty subset of a finite set of linearly independent vectors are linearly independent.

1.1. Vector Spaces over Division Rings

3

Definition 1.2: Let v, ui, v2, • • •, vr be row vectors in Z)(n). If there are elements a1? a 2 , • • ■, a r G JD such that v = ai^i + a 2 u 2 H

f- arvr,

then we say that v is a linear combination of vi, u2, • • •, v r with coefficients «i, a 2 , • • •, a r in -D.

D

Let vi, u 2 , • • •, vr be r row vectors in 2 ? ^ , denote by < v x , u2, • • •, v r > the set of all linear combinations of v\, u2, • • •, vr which coefficients in D. Clearly, if u, v G < Vi, t>2, • • •, vr > and a £ D, then u + u G f l and av G 2). More generally, we have Definition 1.3: A nonempty set V of D^ with the property that u , u G V and a G D imply u + v G V and av G D is called a subspace of 2}(n). If V consists only of the zero vector, then V is called the zero subspace and denoted by < 0 >; otherwise, V is called a nonzero subspace. □ For example, < vi, v2, • • •, v r > is a subspace of D^ a subspace of D^.

and D^

itself is also

Definition 1.4: Let V be a subspace of D ' n ' . If there are row vectors vi, v2, • • •, vr such that V = < v\, u2, * • •, vr >, then they are said to form a set of generators of V and V is said to be spanned by them. Furthermore, if v1? u2?* • •, v r are linearly independent, then they are said to form a basis of V.

□ For example, ei, e 2 , • • •, e n form a basis of 2)(n). We want to prove that every subspace of D^ has a basis and the numbers of elements in different bases of a subspace are the same and < n. We proceed as follows. L e m m a 1.1: Letvi, v 2 , •••, vr be row vectors in D^. Thenwi, t>2, •••, vr are linearly independent if and only if any row vector v G < vi, v2, • • •, vr > can be expressed uniquely as a linear combination of t>i, v 2 , • • •, vr. Proof: Assume that vi, v2, • • •, vr are linearly independent. Let v be any vector of < Vi, u2, • • •, vr >. If v has two representations as linear combinations of vi, v2, • • •, wr, say u = a\V\ + a2V2 + • • • + a r v r

Chapter 1. Linear Algebra over Division Rings

4 and

vr, v = 62v2 H+ • -• • +h b£Tw = biVi 61^2 + b where a t , 6t G 2? (z = 1,2, • • • , r ) , then subtracting the second expression from the first one, we obtain {arr--0 = (en (ai -- h)vi 6i)vi + (a 2 - 62)v2 + • -• +h (a

br)vr.

Since ui, v2, ***> v r are linearly independent, we have at- = 6t-

(z =

1, 2, • ■ ■ , r ) .

Conversely, suppose that any row vector v G < Vi, v 2 , • • •, vr > can be expressed uniquely as a linear combination of vi, v 2 , • • •, vr. Let ai, a 2 , • • •, ar £ D be such that aiVi • +Gr^V arvr ==0.0. ai^i + a22vv22 + • •■■• +

We also have 0vx1 + 0v2 + •"'■ + • +0v0v r r==0.0. By the uniqueness of the zero vector 0 as a linear combination of v\, v2, • • •, v r , we have a t = 0 (z = 1, 2, • • •, r). Therefore vi, v2, • • •, u r are linearly independent. D L e m m a 1.2: If ui, w2, • • •, us G < i>i, v2, • • -, vr > and u> G < ui, u 2 , • • •, u 5 >, then w G < vi, v2, • • •, v r >. Proof: From

rr

(t " , 5*' s)) (* == l,*> 22,> •••*

U

i = ]Cfl«*V* 4=1 A:=l

and s

w ii = = ll

where alfc, 6t- € 25 (i = 1, 2, • • •, 3; k = 1, 2, • • •, r), we deduce s s

r

r

s

kVk = U> = X) b* 2 3 a^iiJbVJb ^2(YJ W

*i=l =1

k=l

k=l

h a

i*)v*. i ik)Vk-

i'=l

D

Theorem 1.3 (Replacement Theorem): Let ?ii, u2, •••, u s be 5 linearly independent elements of the subspace < vi, v 2 , • • •, vr >, then there exist 5

1.1. Vector Spaces over Division Rings

5

elements vt-a, vt-2, • • •, vis (1 < z'i < z2 < • • • < is < r) in {^i, v 2 , • * *, vr} such that if we replace them by u i , W2, • • •, us in {vi, v2, • • •, ^ r } , the resulting set is also a system of generators of < u1? u2, • • •, v r > . In particular, s < r. Proof: We prove this theorem by induction on 5. If 5 = 0, the theorem holds trivially. Assume now that the theorem holds for 5 — 1. Then there exist 6 — 1 elements t ^ , ut-2, • • •, V{s_l (1 < ix < i2 < • • • < zs_i < r) in {vi, v2, • • •, v r } such that if they are replaced by ui, u 2 , • • •, w s -i then the resulting set {ui, u 2 , * • *, ^ s - i , ut-tf, v ts+1 , • • •, vt-r} is also a system of generators of < Vi, v 2 , • • •, v r > , where z'i, permutation of 1, 2, • • •, r. Since ws G< vi, ^25 • • •, t>r > v1,v22,,-',v-r ' < vi,v

> = < u u ,-

an

d

• •, W ,u—1 ,, ^t' s.uv,isf,vis+1

, Vr > = < Uui , 2 W2,- •

S

?s

- ir,v> ir ,> , •-■,v

5 " "

s+ 1

we have = aa^ui + aa22uu2 2 + -\ • •■• +h a ss _it/ _ i t / s _i _i + + assV{ v;s5 + + a a ss+1 vis+1 + • •• ■ uus = i^i + + iV z - 5 + 1 -\

(1.1) +\- aarrvvirir,, (1.1)

where ai, a2, • • •, a s _i, a s , a s + i , • • •, ar G D. By hypothesis t/i, U2, • • •, us are linearly independent over D, thus a s ,a s +i, • • •, ar cannot all be 0. Without loss of generality we assume that the subscripts zs, z s +i, • • •, ir have been so chosen that as ^ 0. Then •— Vi - a / a aiUi i ^ i -—• •• •• •——asa~asa-iu asajusus-a— a~ • a• s • —arV{ a~r. arV{r. s-iu s-is-i ++ s+iVis+1• •— s a s+1vais+1 is s = = —aj V

It follows that V{3 G < wi, • • •, w s _i, u s , ut-a+1, • • •, V{r > and consequently <

Wl,""

, WS —1 ? ^ * s ? ^'5+1 5 *. ',u l V

> C < ui,--

• , Us—1, U s , ^z' s+1 5 ' '

• , ^tv > •

On the other hand it follows from (1.1) that us G < t*i, u 2 , • • •, u s - i , ^t-a, u

t a+ i?

,,,

, U t r > and therefore

< u iUl," , - - - ,, uWs __ ii ,, uU s ,, ^^ i5 + 1 , - - --,v, ir^ r > >C C<
s

s

s

5+15

t a

a+1,

,v

' '■ • , Vi ir r

>> .•

Hence >v < vuv2,- 'r>,V

r

= < uUuui 2,,-U , "- ' ,u ,via+1 ,vir >> 5 V's-l') ) Vi,- , • •• • •, Vir 8-i,viaV{ = < U-L , • • •, w -i, w , v; , • • vir .> . • , U —1, 1/ , ^z' • * •, v = < w i , w, w , s s s+1 ir•, > 2 2

> = <

3

2

s

s

s+1

s+1 ?

D

6

Chapter 1. Linear Algebra over Division Rings

From the Replacement Theorem we deduce Proposition 1.4: Any nonzero subspace of D^ has a basis whose number of elements is < n and any two bases of the subspace contain the same number of elements. Proof: Let V be a nonzero subspace of D^n\ Choose any nonzero element v\ in V. If V = < vi >, then V\ forms a basis of V. If V ^ < V\ > , then there is an element v2 in V such that v2 # < v\ >. Thus < Vi,v2 > Q V and vx, v2 are linearly independent. If V = < v\,v2 >, then Wi, v2 form a basis of V. If V =fi < v^v2 >, then there is an element v 3 in V such that v3 £ < vi,v2 >. Thus < vi,t>2,^3 > Q V a n d v i^ v 2, ^3 are linearly independent. By the Replacement Theorem the number of elements in any set of linearly independent vectors of D^ = < ei, e 2 , • • •, e n > is < n. Thus proceeding in the above way, we will arrive at a set of linearly independent elements vi, v2, • • •, vr in V such that V = < t>i, v2, • • •, vr > and r < n. Hence i>i, v2, • • •, v r form a basis of V. This proves that V has a basis whose number of elements is < n. Let {vi,^2,-• • ,v r } and {1/1,1*2, • • •,u s } be two bases of V, then V = < ^ i ^ 2 , • • •, vr > = < tii, ^2, • • •, us > . By the Replacement Theorem we have r < s and s < r. Therefore r = s. □ Based on Proposition 1.4 we give the following definition. Definition 1.5: The number of elements in any basis of a nonzero subspace V of D^ is called the dimension of V and is denoted by dim V. In particular, dim Z)(n) = n. The dimension of the zero subspace of D^ is defined to be 0, i.e., d i m < 0 > = 0. D Corollary 1.5: The dimension of < v1? v2, • • •, vr > is the maximal number of linear independent elements among v1? v2j • • •, vr. D Proposition 1.6: Let V and W be subspaces of D^ of dimensions / and m, respectively. Assume that V CW. Then / < m and to any basis of V we can add m — / elements of W such that the resulting set is a basis of W. In particular, if V C W and / = m, then V — W. Proof: It follows immediately from Theorem 1.3.

□

1.1. Vector Spaces over Division Rings Now let us study the intersection and sum of subspaces of

7 D^.

Proposition 1.7: Let V\ and V2 be two subspaces of D^n\ Then the set row vectors belonging to both V\ and V2 is a subspace of D^ and the set row vectors which can be written as sums of a vector of V\ and a vector V2 is also a subspace of D^.

of of of □

The proof is easy and hence omitted. Definition 1.6: Let Vi and V2 be two subspaces of D^n\ The subspace of Z)(n) which consists of those row vectors belonging to both V\ and V2 is called the intersection of Vi and V2 and denoted by V\ f) V2. The subspace of Z)(n) consisting of those row vectors which can be written as sums of a vector of V\ and a vector of V2 is called the sum of Vi and V2 and denoted by Vi + V2. □ The following properties of the intersection and sum of subspaces are imme­ diate. Let V, Vi, V2, and V3 be subspaces of D^n\ then

vnv = v, v + v = V; v1nv2 = v2nVi, vl + v2 = v2 + Vi; vi n {v2 n v3) = (Vi n v2) n v3, v1 + (v2 + v3) = (vi + v2) + v3; vi n (Vi + v2) = Vi, vi + (Vi n v2) = vi. More importantly, we have Proposition 1.8 (Dimension Formula): Let Vi and V2 be subspaces of D^n\ then dim VL + dim V2 = dim (Vi C\V2) + dim (Vi + V2). Proof: Let dim(Vi fl V2) = d and ui, v2, • • •, ^ be a basis of Vi D V2. Clearly Vi fl V2 C Vi and Vi n V2 C F 2 . Let dim Vi = s and dim V2 = t. By Proposition 1.6, d < s, d < t, there exist elements ui, u 2 , • • •, u3-d in Vi such that v\, v2, • •, vj, u 1? w2, • • •, u5_d form a basis of Vi, and there exist elements wi, w2, • • •, wt-d in V2 such that vi, v2, • • •, Vd, u>i, w2, • • •, wt-d form a basis of V2. Clearly V1 + V2 =
• • •, vd, ui, • • •, u3-d, wu • • •, w*_d > .

If we can prove that vi, • • •, vj, «i, • • •, wa_d, itfi, • • •, wt-d are linearly inde­ pendent, then our theorem will be proved. Suppose that there are elements

8

Chapter 1. Linear Algebra over Division Rings

0i) * * •) ad, h, ' • •, ^5-d, ci, * • ■? Ct-d G D such that aiui H

h a^v* + ftitii H

h bs-dus-d

+ ciwi H

h adVd + 6iMi H

h bs-dus-d

= —(ciWi H

h ct-dwt-d

- 0.

Then ai^i H

h

ct-dwt-d).

The left-hand side of the above equality is an element of Vi and the righthand side is an element of V2, hence it is an element of Vi fl V2. It follows that 61 = • • • = bs-d = 0 and c\ = • • • = ct-d = 0 and consequently are ai = ••• = ad = 0. Therefore t>i, • • •, v^, uu • •, u5_d, Wi, • • •, wt-d linearly independent. D From the dimension formula we deduce Proposition 1.9 (Modular Law): Let Vi, V2, and V3 be subspaces of and assume that Vi C V3. Then

D^

K + (v2nv3) = {K + v2) nv3. Proof: From Vi C Vi + V2 and Vi C V3 we deduce Vi C (Vi + V2) n V3. Clearly, V2 0 F 3 C (Vi + V2) n V3. Therefore Vi + (V2 H F3) C (Vi + V2) n V3. If we can prove

dim(Vi + (v2 n y3)) = dim((Vi + v2) n v3), then our proposition follows immediately from Proposition 1.6. By Propo­ sition 1.8, we have dim(Vi '+ (V2 H V3)) = dim Vi + dim( V2 n V3) - dim(Vi f)V2n V3) = dim Vi + dim(y 2 H V3) - dim(Vi fl V2) (as Vi C V3) = dim Vi + dim V2 + dim V3 — dim( V2 + V3) - ( d i m Vi + dim V2 - dim(Vi + V2)) = dim V3 + dim(Vi + V2) - dim(V2 + V3) and dim((Vi + V2) H V3) = dim(Vi + V2) + dim V3 - dim(Vi + V2 + V3) = dim V3 + dim(Vi + V2) - dim(T/2 + V3) (as Vi C V&). D

1.1. Vector Spaces over Division Rings

9

We remark that if the condition v\ C V3 does not hold, then the conclusion of Proposition 1.9 may not be true. It is not difficult to give counter-examples. Proposition 1.10: For any subspace V of D^ subspace W such that D^

= V +W

and

VHW

=

there exists at least a

<0>.

Proof: Let dim V = r and v l5 u2, • • •, vT be a basis of V. By Proposition 1.6 there are row vectors i/i, u 2 , • • *, un-r such that vi, v 2 , • • •, ^r, ^ i , ^2, • • •, un-r form a basis of D^n\ Let W = < i/i, u 2 , • • ■, u n -r >, then

£>(") = y + w and v n w = < o > . Correspondingly, we have the n-dimensional column vector space (or right vector space) over JD, which consists of all n-dimensional column vectors

.

,

X{, e D, i = 1, 2, • • •, n,

\ Zn /

over D with addition and scalar multiplication defined by / 2/1 \ 2/2

^2

/ ^1 + y\ \ I ^ 2 + 2/2

+ \ ^n /

V 2/n /

\ Xn + Vn /

and ( X\X

\

X2X

\ Xn /

\ xnx I

respectively, where #, xi, # 2 , • • •, xn, j/i, y 2 , • • •, yn G D. We use the symbol *D(n) to denote the n-dimensional column vector space over D and lx to denote a vector in it, where x is an n-dimensional row vector over D. We have analogous results for *Z?(n) but they will not be repeated here.

10

Chapter 1. Linear Algebra over Division Rings

Finally, let us give the definition of an (abstract) vector space over a division ring. Definition 1.7: Let D be a division ring and V a nonempty set. Assume that to any two elements v and w of V there corresponds uniquely an element of V, which is called the sum of v and w and denoted by v + w. This correspondence is called the addition of V. Next assume that to any element A £ D and v G V there corresponds uniquely an element of V which is called the scalar product of v by A from the left and denoted by Xv. This correspondence is called the scalar multiplication of V. We say that V is a left vector space over D if the following manipulation rules I and H hold in V. I

V is an abelian group with respect to the addition.

H For any A, fi £ D and v, w € V, we have (i) (ii) (iii) (iv)

X(v + w) = Xv + Xw, (A + fi)v = Xv + fiv, (\p)v = A(/*v), lu = u.

We call the elements of V vectors, the zero element of the abelian group with respect to addition the zero vector, denoted by 0, and the elements of D scalars. Furthermore, besides I and H, assume that the following also holds: 1H There are n vectors ei, e 2 , • • •, e n of V such that any vector v can be expressed uniquely as their linear combination v = cici + c 2 e 2 + • • • + c n e n , with coefficients Ci, c 2 , • • •, cn in D. Then V is called a finite dimensional left vector space over D, n its dimen­ sion, and {ei, e 2 , • • •, e n } a basis of V. □ is an example of a finite dimensional left vector space over D of dimen­ sion n.

1.2. Matri ces over Di vision Rings

11

Two left vector spaces over the same division ring are said to be isomorphic, if there is a bijective map from one to the other which preserves the addition and scalar multiplication. Parallel to the discussion of D^n\ for finite dimensional left vector space V over a division ring D we can repeat all the definitions and propositions given in this section, but the details will be omitted. Similarly, we can also define finite dimensional right vector spaces over D. When D = F is a field, the map from F^

to

f

F^

I «i \ a2

(ai, a 2 , •••, an)

(1.2)

\ «» / is bijective and preserves the addition and scalar multiplication of vectors. In fact, (ai,a 2 ,- • • , a n ) + (61,62,- • ■, 6n) = («i + &i>a2 + 62, * •• , a n + M / ai + 61 \ a 2 + 62

/ d\ \ a2

V a n + bn }

\ «n /

/ 6l \ 62

+

VK )

and a(ai, a 2 , • • •, a n ) = (a«i, aa 2 , • • •, aan) ( ««i

aa 2

\

/ aia \ a a

2

( «i

\

«2

\ ^ / Thus F^ ^ and F^ can be regarded as isomorphic. Usually we identify and
1.2

f

F^

Matrices over Division Rings

Let Z) again be any division ring, m and n be positive integers, and a^ (i = 1, 2, • • •, m; j = 1, 2, • • •, n) be mn elements of D. We arrange the mn

Chapter 1. Linear Algebra over Division Rings

12

elements a^ of D in a rectangular array /

#11

#12

' '

#ln

#21

#22

• *

#2n

\ #ral

«m2

'*

\

(1.3) /

and call it a matrix with m rows or n columns over the division ring D or simply, a n m x n matrix over D. Capital letters are often used to denote ma­ trices. Denote the above matrix (1.3) by A which is sometimes abbreviated to A ~ (aij)l
U'4)

The row vectors ( # 1 1 , # 1 2 , • * • , # l n ) 5 ( # 2 1 , #22? * * ' , # 2 n )

• " ,

a

( # m l , «m25 ' * *5

mn)

are called the first, second, • • •, m-th row vectors of A, respectively. They are row vectors of D^. The column vectors

I flu )

^ #12

#21

\

/

«22

#ln

\

#2n

?

\ #ml

/

\ #m2

/

\ Q"mn /

are called the first, second, • •, n-th column vectors of A, respectively. They are column vectors of *D'm). We adopt the convention that zeros in matrices are sometimes omitted, i.e., the blanks replace the omitted zeros, if it is clear from the context. For instances, the matrices 0

• ••

#22

* *•

/ #11 #21

\

#ml

#ra2

'

0 ^ #2n

/

#11

#12

#21

#22

0 0

0 0

f #11

#12

#21

#22

and \

Q"mn /

0 0

0

\

0

#33

034

#43

#44

/

may be written as /f

#11

\

#21

'^

Ami

#22

a

m2

'

''

0>2n

Q"mm /

and

{

\ #33

#34

#43

#44

/

1.2. Matrices over Division Rings

13

respectively. Let us introduce the operations on matrices. (i) The addition of matrices. Let A = {
B = (&tj)l
be two m x n matrices over D. Define an m x n matrix over D C = ( c t j ) i < 2 < m i \<j

where Cij = dij + bij,

i = 1,2,- • • ,ra; j = 1,2,- • • ,n.

We call C the swra of A and B , which is denoted by C = A + B. Note that two matrices can be added if and only if they have the same number of rows and the same number of columns. It is easy to prove that all m x n matrices over a division ring D form an abelian group with respect to the above defined addition of matrices. In particular, we have A + B = B + A, (1.5) (A + B) + C = A + (B + C).

(1.6)

The zero element of this abelian group is the matrix whose elements are all O's, which is called a zero matrix and denoted by 0^m,n\ or, in short, 0. If m = n we also write 0 ^ for 0^ m ' n \ The negative of the matrix A = (atj)i
=A +

{-B),

where A and B are both m x n matrices. We may regard the n-dimensional row vectors over D as 1 x n matrices over D. The addition of n-dimensional row vectors over D is the addition of 1 x n

Chapter 1. Linear Algebra over Division Rings

14

matrices over D. Similarly, the addition of m-dimensional column vectors over D is the addition of m x 1 matrices over D. (ii) The scalar multiplication of a matrix over D by an element of D. Let (1.4) A =

{aij)l
be an m x n matrix over D and c £ D. Define a matrix C

=

( c t'j)l

where Cij = caij,

i = 1,2, • • •, m; j = 1,2, • • •, n.

We call the elements of D scalars and C the scalar product of A by the scalar c in D. We may write the m x n matrices over D as mn-dimensional row vectors over D in the following way. For the matrix A in (1.4) we write (an 5 ^12,

, # i n , C&21, ^ 2 2 j

> ^2n5

* ?a

m

j ,
m n

)• (1-7) Then the scalar multiplication of A by a scalar c in D corresponds to the scalar multiplication of the mn-dimensional row vector (1.7) by c. Also, the addition of two m x n matrices over D corresponds to the addition of their corresponding mn-dimensional row vectors. Thus the set of m x n matrices over D with addition and scalar multiplication defined above becomes the mn-dimensional left vector space over D with respect to addition and scalar multiplication. Therefore we have the manipulation rules (1.5), (1.6), and c(A + B) = cA + cB, (c + d)A = cA + dA, (cd)A = c(dA), 1A = A. Moreover, denote by Eij the m x n matrix whose element at position (z, j) is 1 and all other elements are O's, then the mxn matrix A in (1.4) can be uniquely expressed as m

n

t=l j=l

1.2. Matrices over Division Rings

15

Hence {Eij \ i = 1,2, •• - ,m\j = 1,2, •••,ra} is a basis of this left vector space and this left vector space is of dimension run. Similarly we define the scalar multiplication of a matrix A = (o>ij)i
B =

(&t'j)l
Define an m x / matrix C ~

{cij)l
where n

Cij = $ 3 ° * * ^ '

i = 1,2, • • • ,m; j = 1,2, • • •,/.

A;=l

We call C the product of A and 5 , denoted by C = AB. Note that only when the number of columns of the first matrix is equal to the number of rows of the second matrix the two matrices can be multiplied. Multiplication of two matrices are performed according to the rule of "row by column", that is, multiplying the elements of the z-th row of the first matrix and the corresponding elements of the j-th column of the second matrix and summing up the products yields the element at position (i,j) of the product matrix.

Chapter 1. Linear Algebra over Division Rings

16

It can be easily proved that the matrix multiplication is associative and that it is also distributive with respect to addition, that is, (AB)C =

A(BC),

A(B + C) = AB + AC, (A + B)C = AC + BC, as long as the multiplication and addition in the above equations are defined. But even if when A and B are n x n matrices, it is possible that AB ^ BA. For example,

i i wo

i wi

0 1) \ 1 0)

o 1W1 1 0

i \ 1 0

i

iWo 0 1

1 1 1

and 1 1 1 0 I

T

0 1 \ 1 1

It can also be proved that the following manipulation rules hold: c{AB) =

(cA)B,

(AB)d=A{Bd). Let A be an n x n matrix. We call the set of positions (1,1), (2,2), • • •, (n,n) oi A the principal diagonal. If all elements of A except those on the principal diagonal are O's, then A is called a diagonal matrix. The n x n diagonal matrix whose elements situated at the principal diagonal are ai, ^2) " '•) an in succession will be denoted by [a1,a2,---,an] for simplicity. We denote by 1^

the n x n diagonal matrix [i,i,---,i]

and call it the n x n identity matrix. Sometimes we abbreviate 1^ Clearly for any m x n matrix A, we have AJT = A,

I^A

= A.

to I.

1.2. Matrices over Division Rings

17

(iv) The transpose of a matrix. By interchanging rows and columns of a matrix A, we obtain a matrix which is called the transpose of A, denoted by l A. Let A be the m x n matrix (1.4), then *A is the n x m matrix \\

n / O #11

#21

#ml ** ' ' ** #ml

#12

#22

** ' *' ' #ra2 #m2

\ ##llnn

#2n

' * " @"mn 0>mn) )

It can be easily proved that

'{'A) --=A,A, *('A) t

l

(A ++ 5B)) = A + *B, <(A = *A+ *B, f HcA) ( c i ) ===ccU. A

f

Notice that since the multiplication of D is not necessarily commutative, we do not have t{AB) = *B tA in general. However, if r is an anti-automorphism of D and we denote by Ar the matrix obtained from A by applying r to all the entries of A, then T r [(AB)T]== \BTy(A \Br)\A %ABY\ ). ).

t

Let us define the submatrices of a matrix. Let A = A=

( # z j ) l < ii<m < m,l<j
be an m x n matrix over a division ring D. Let z'i, z*2, • • •, ir be indices such that 1 < z'i < i2 < • - - < ir < Tn and let j i , j 2 , • • •, js be indices such that 1 < ji < J2 < ''' < js < TI. Then the matrix ttzijj /f a{ 1j1

1j2 i\32

a

a

a

a

i23\ t2Jl

t2J2 i2J2

a

a

irjl

a

\ \

acLi

irjl

a

irJ2

irJ2

'''

aa

iij hjss

\

a

'''

i2Js i2Js

a

" ' ' '

#*Vjs / a

irjs

is called a submatrix of A. Let us introduce the block form of a matrix.

/

18

Chapter 1. Linear Algebra over Division Rings

Let A be an m x n matrix over a division ring D and assume that m is decomposed into a sum of s positive integers m i , m 2 , • • •, ms and that n is decomposed into a sum of t positive integers rai, n 2 , • • •, nt: h mra55,, n = m = mi + m 2 + • • + —nin\++n,2 n2 + -• •—++ntn. t. Denoted by Aij (1 < i < s, 1 < j < t) the submatrix of A obtained from A by deleting all rows of A except rows

mi+m 2 + ••+m;_i + l, mi+m 2 +

hm,_i+2, •••, mi+m 2 +

hmt_i+m;

and deleting all columns of A except columns ni+n2 + hrij-i -• •, n1i-\+rnn22 + nj_i ++ 1, l, nnii++nn22 ++ - . +b n^ j. _1i++ 2 , •••, + - -+nt-rij^+rij. ^1+^2 +' •+ j-1+nj. Then AtJ- is an mt- x nj submatrix of A. More precisely, let m(- = m i + m 2 + • • • + m t _i and n'j = n\ + n 2 + • * * + ^ j - i , then // = A^ Aij — I

\

^mj+l,nj+l «m;+l,n:+l

aa

m(+l,nJ+2 m(+l,nJ+2

' ' '*

«m;+2,n^+l

«m;+2,nJ+2

* '

a

, m[+l flmj+l.nj+n,1n j+nj

m'i+2yj+nj

. a

mJ+mi,nJ+l

\\

a

.

fl

mJ+mi,n'+2

a

'

mJ+mj,n'+nj

J

A can be written in the following block form

A =

/(AnAn A22i i A

A A1212 ■-" ■ AAltl

A2222 ■• A

t

\\

•• • AA2t2t •>

V\AA As2 s2 ■•■• •• AAst st ) J A sl A where A tJ (i = 1,2, • • •, r; j = 1,2, • • •, s) are called the 6/ocfcs of A. Next let B be an n x I matrix over D and assume that / is decomposed into a sum of r positive integers ■lT. l = h + li l2 ++ - ~-lr.

Denote by Bjk (1 < j < t, 1 < k < r) the submatrix of B obtained from B by deleting all rows of B except rows nj_i ++ l,l, ^1+^2 ni+n2 + + * • +hrij_i ni+n2 + * • + h n^•-1 j _ i ++22, , •••, n1+n2 2 + - • +h nn^j _i i++nn.j ^1+^2 •••, ni+n

1.2. Matrices over Division Rings

19

and deleting all columns of B except columns

h + h + '" + h-i + 1, h + h + ~' + h-i +2, • • •, h + l2 + "' + h-i + /*. Then Bjk is an rij x /* submatrix of B and B can be written as I Bn

B12 - • • Blr\

B21

B22

'''

\ Bn

Bt2

• • • Btr )

We have the multiplication formula of matrices in block form: AB =

(Cik)l
where t

Cik =

y^AqBtk, i=i

in which the multiplication and addition involved are the matrix multiplica­ tion and matrix addition, respectively. There is an addition formula and a scalar multiplication formula for matrices in block form as well. Since they are much simpler than the multiplication formula, we do not write them down here. Now let us introduce invertible matrices. Let A be an n x n matrix over a division ring D. If there is an n x n matrix B over D which satisfies AB = BA = / , when / is the n x n identity matrix, then A is called an invertible matrix and B is called an inverse matrix of A. If B\ is another inverse matrix of A, that is AB1 = BXA = 7, then B = BI = B(AB1)

= {BA)B1

= IBl = Bx.

20

Chapter 1. Linear Algebra over Division Rings

Hence the inverse matrix of an invertible matrix is unique. We denote the inverse matrix of an invertible matrix A by A - 1 . It is clear that A - 1 is also an invertible matrix and (A-1)-1 = A. Also, if A and B are both n x n invertible matrices, then AB is also an invertible matrix and {AB)-1 = B~lA-\ It can be easily proved that the set of all n x n invertible matrices over a division ring D forms a group with respect to the matrix multiplication, which is called the general linear group of degree n over D and is denoted by GLn(D). The identity element of GLn(D) is the identity matrix and the inverse element of A is A" 1 . We define a n n x n matrix A over D to be left invertible if there is an n x n matrix B over D such that BA = I^n\ and B is called a left inverse of A. Similarly, we define A to be right invertible if there is an n x n matrix C over D such that AC = I^n\ and C is called a right inverse of A. Let A be the m x n matrix (1.4). Denote the first, second, • • •, m-th row vectors of A by ai, 02, • • •, a m5 respectively. That is, di — ( f l t l , « t 2 , * ' ' , O t n ) ,

Z=

1 , 2 , • • • , 771.

Then we have the following criterion for left invertibility of matrices. Proposition 1.11: An nxn matrix A over a division ring D is left invertible if and only if the n rows of A are linearly independent. Proof: Denote the n rows of A by ai, a 2 , • • •, an in succession. Then

a

H -- ■ At first, suppose that A is left invertible, i.e., there is an n x n matrix B over D such that S A = I^n\ It is clear that the n rows of / M , i.e., ei, e 2 , • • •, e n , are linear combinations of the n rows «i, 02, •••, a n o f A. Then

1.2. Matrices over Division Rings

21

j)(n) = < e i , e 2 , - " >en > C < aua2,-" ,an >. But aua2,", a n G £> (n \ hence £Hn) = < a i , a 2 , • • • , a n >. If ai, a 2 , • • •, an are linearly dependent, then dimZ)( n ) < n, which is a contradition. Therefore ai, a 2 , • • •, «n are linearly independent. Conversely, suppose that au a 2 , •••, an are linearly independent. Then < o>ij a>2->'' *, dn > — D^n\ Each et- (1 < z < n) can be expressed as a linear combinations of a1? a 2 , • • •, an. That is, the rows of 1^ can be expressed as linear combinations of the rows of A. Hence there is an n x n matrix B such that 7(n) = BA. Therefore A is left invertible. □ We can also state a similar criterion for right invertibility of matrices in terms of the column vectors. Proposition 1.12: Let Abe n x n matrix over a division ring D. Then A is left-invertible if and only if A is right invertible. If A is left invertible (or right invertible) then the left inverses and right inverses of A are all equal and hence A is invertible. Proof: Let A be left invertible. Then there is an n x n matrix B such that BA = I< = r and that A is of column rank s if dim < 61? 62, • • •, bn > = s. In the following we shall prove that the row rank of a matrix is equal to its

Chapter 1. Linear Algebra over Division Rings

22

column rank. At first, we shall see that by means of the row rank Proposition 1.11 can be restated as follows. Proposition 1.13: An n x n matrix A over a division ring D is invertible if and only if its row rank is n. □ Moreover, we have the following propositions. Proposition 1.14: (i) Let A be an m x n matrix of row rank r over a division ring D and B be an n x I matrix of row rank n over D. Then AB is also of row rank r. (ii) Let A be an m x n matrix of column rank n over D and B be an n x I matrix of column rank s over D. Then AB is also of column rank s. Proof: It is sufficient to prove (i) only. Since B is an n x / matrix of row rank n, the n rows of B are linearly independent. Let A\ be an s x n matrix obtained from A by deleting m — s rows. Let 6 G £>W. Then bAxB = 0 if and only if bA\ = 0. It follows that the maximum number r of linearly independent rows of A is also the maximal number of linearly independent rows of AB. □ Proposition 1.15: Let A be an m x n matrix of row rank m over a division ring D, then m
U) is invertible. Proof: Denote the rows of A by Vi, v2, • • •, vm in succession. Since they are linearly independent, we have m < n. By Proposition 1.6 we can add n — m row vectors v m +i, vm+2> • • •, v n to them such that Vl,V2,"

form a basis of D^.

' , V v m + l ) U m + 2 ) ' * ',vn

Let /f vvm+i m+i \\ Vm+2 Vm

B B=\=

+>

\

vn

. )

1.2. Matrices over Division Rings

23

Then ' A B is invertible.

□

Definition 1.9: Let A and B be both m x n matrices over a division ring D. They are said to be equivalent if there is an m x m invertible matrix P and an n x n invertible matrix Q such that A = PBQ. □ Proposition 1.16: Equivalent matrices have the same row rank and the same column rank. Proof: Let A and B be equivalent m x n matrices over a division ring D. That is, there is an m x m invertible matrix P and an n x n invertible matrix Q such that A = PBQ. Then P " 1 A = BQ. By Proposition 1.14 B and BQ have the same row rank. Clearly A and P~lA have the same row rank. Therefore A and B have the same row rank. Similarly we can prove that A and B have the same column rank.

□

Proposition 1.17: Any m x n matrix of row rank r over a division ring is equivalent to I(r) r\(m—r,n—r)

Proof: Let A be an m x n matrix of row rank r over a division ring D. An mxm matrix P is called a permutation matrix if every row and every column of P has only one 1 and m — 1 O's. Clearly lPP — / , hence permutation matrices are invertible matrices. Moreover, PA is a matrix obtained from A by permuting its rows. Since A is of row rank r, we may choose an mxm permutation matrix P such that the first r row of PA are linearly independent, then the last m — r rows of PA are linear combinations of the first r rows. Write

PA

v2 \Vrn/

Chapter 1. Linear Algebra over Division Rings

24 then

r

i = r + l,r + 2,-- • ,m.

i=i

Let

Band

= (bij)r +l
f /<'>

P ~ (

\

I{T)

/( m - r ) )J '

-B

Then Pi is an m x m invertible matrix and

PtPA =

(

where

# =

R

\

Q(m-r,n) J >

f VVl ) 2

\VT • _ ._

) -i r

-1

Since R is of row rank r, by Proposition 1.15 there is an (n — r) x r matrix D such that (' R R\

Q=

[»)

is an n x n invertible matrix. Then

( /W I

Q(m-r,n-r)

\^

J<5 =

(

R\

loj-

Hence

PiPAQ-1 = (

/(r)

Q(m-r,n-r)

^1

• D

Corollary 1.18: The row rank and column rank of any m x n matrix over a division ring are equal. D Definition 1.10: Let A be an m x n matrix over a division ring. Then the common value of the row rank and column rank of A is called the rank of A, which is denoted by rank A. D

1.2. Matrices over Division Rings

25

C o r o l l a r y 1.19: Let A and B be m x n matrices over D. Then A and B are equivalent if and only if they have the same rank. □ P r o p o s i t i o n 1.20: Let A be an m x n matrix of rank r over a division ring D. Then A has r x r submatrix of rank r and any s x s submatrix of A is not invertible if s > r. Proof: Since A is of rank r, A is of row rank r and there are r linearly independent rows of A. Denote the submatrix formed by these r linearly independent rows of A by A\. Then A\ is an r x n matrix of rank r. Hence A\ is also of column rank r and there are r linearly independent columns of A\. Denote the submatrix formed by these r linearly independent columns of Ai by A2. Then A2 is an r x r submatrix of A\ and also of A. A 2 is of column rank r and, hence, rank r. Let B be an s x s submatrix of A and s > r. If B is invertible, by Proposition 1.11 the s rows of B are linearly independent. It follows that the s rows of A at which the s rows of B are situated are also linearly independent. Then rank A > 5, which is a contradiction. □ Now we give two propositions on the bounds of the ranks of the sum and product of two matrices. P r o p o s i t i o n 1.21: Let A and B both be m x n matrices over a division ring D. Then rank(A + B) < rank A + rank B. Proof: By Proposition 1.16 and Corollary 1.18 the rank of a matrix is invariant under the equivalence transformation X 1—* PXQ, where X is an m x n matrix, P G GLm(D), and Q G GLn(D). By Proposition 1.17, without loss of generality we can assume that A =

JM

\ r\(m—r,n—r)

where r = rank A. Let

*-(£)•

Ii

26

Chapter 1. Linear Algebra over Division Rings

where B\ is an r x n matrix and B2 is (m — r) x n. Then rank(A + B) < rank((J( r ) 0) + Bx) + rank B2 < rank A + rank 5 .

□ Proposition 1.22: Let A be an / x m matrix over a division ring D, B be an m x n matrix over D. Then rank A + rank B — m < rank AJ3 < min{rank A, rank B}. Proof: (i) For simplicity let C = AB, rA = rank A, r# = rank 5 , and rc = r a n k C There is an M G GLm(D) and TV € GLn(D) such that

M

B

/ j(r*)

= {

\

o)N-

Let C* = CN~X and A* = AM, then rankC* = rCj rank A* = rA, and A

*(^

o) = ^

Write A* = (AJ AJ), where A\ is / x r^ and A2 is / x (m — r ^ ) , then C* = (AJ 0). Since A* has r^ linearly independent columns, A\ has at least TA — {m — TB) linearly independent columns. Therefore rc > r^ + r^ — m. (ii) From C = AB we know that the rows of C are linear combinations of the rows of B. Therefore the row rank of C is less than or equal to the row rank of i?, i.e., rc < rs> Similarly, the columns of C are linear combinations of the columns of A. Hence rc < rAD Definition 1.11: Two m x n matrices A and B over a division ring D are said to be row equivalent if there is an m x m invertible matrix P such that A = PB. As in the case when D is a field we can prove that any m x n matrix over a division ring D is row equivalent to an echelon matrix, called its echelon normal form.

1.3. Matrix Representations

27

of Subspaces

Proposition 1.23: Let A be an m x n matrix over a division ring D, then A is row equivalent to a matrix of the following form:

* 0 * 0 1 *

* * *

0 * 0 * 0 *

* *

0 0 0 0 0 0

0 0 0 0 0 0

0 0

1 * 0 0

* 0

0 0 0

0 0 0

0

0 0

0

/0 •••0 1 * 0 ••• 0 0 0 0 ••• 0 0 0

* 0 * 0 1 * 0 0 0

0 0 0 0 0 0 0 0 0

* o *

where * denotes some element of F. Further, let rank A = r, then all the elements below the r-th row of the above matrix are O's, the first nonzero element of each of the first r rows, from left to right, is 1, and the r l's belong to r different columns. Thus all the elements to the left of each of these l's in the same row are O's, all the other elements in the same column of each of them are also O's, and all the elements below and to the left of each of them are all O's, too. Hence if the first nonzero element 1 of the i-th row (1 < i < r) is located at column &;, then 1 < ki < &2 < * * * < K < n. □ We omit the proof the above proposition; interested readers may consult Wan 1992b, Theorem 5.27.

1.3

Matrix Representations of Subspaces

Now let P be an m-dimensional subspace of D^ and Vi, i^, • • •, vm be a basis of P. Then ^i, ^2, •••, vm are vectors in D^n\ We usually use the m x n matrix V2

\ vm /

28

Chapter 1. Linear Algebra over Division Rings

to represent the subspace P and write

(

Vl

\

\VmJ i.e., we use the same letter P to denote a matrix which represents the subspace P . We call the matrix P a matrix representation of the subspace P. It should be noted that a matrix representing an m-dimensional subspace is an 77i x n matrix of rank m. Of course, two m x n matrices P and Q both of rank m represent the same ra-dimensional subspace, if and only if there is an m x m invertible matrix A such that P = AQ. Elements of GLn(D) can be regarded as linear transformations of D^n\ Let T G GLn(D), then T carries (or transforms) the vector (xi,x2, • • • ,xn) of D^ into (a?i, x2, • • •, xn)T. That is, we have a map £> x GLn(D) —+ £>M ((xi, x2, • • •, z n ) , T) i—► (zi, x2, • • •, £ n )T. We also say that this is an action of GLn(D) on Z ) ^ , or GLn(D) acts on Z>(n). This action induces an action of GLn(D) on the set of m x n matrices over D and also on the set of m-dimensional subspaces of D^: if T G GLn(D) and P is an m x n matrix over D or an m-dimensional subspace of D^n\ then T carries P into P T . In fact, we have Proposition 1.24: If T G GLn(D) and P is an m x n matrix of rank m over /?, then P T is also an m x n matrix of rank m. Furthermore, if P and Q are both m x n matrices of rank m over Z), then there is an element T G GLn(D) such that P = QT. Proof: The first statement follows from Proposition 1.14. Now we prove the second statement. Let P and Q both be m x n matrices of rank m. By Proposition 1.15 there are (n — m) x n matrices B and C such that both

(5) - (?)

1.4. Systems of Linear

Equations

29

are n x n invertible matrices. Let

T = then T G GLn(D)

(?)" ' ( * )

and P = QT.

□

Corollary 1.25: The set of all non-zero vectors of D^ forms an orbit, i.e., a transitive set, under GLn(D). The zero vector (0,0, • • •, 0) is left fixed by every element of GLn(D). □ Proposition 1.26: The set of all m-dimensional subspaces of D^ an orbit under GLn(D). Proof: Use matrix representations of subspaces of D^n\ Proposition 1.27: The number of orbits of subspaces of D^

1.4

forms

□ is n + 1. □

Systems of Linear Equations

Let D be a division ring, a tJ (i = 1,2, • • • , r a ; j = 1,2, • • • ,n) and 6t- (i = 1,2, • • • ,m) be elements of D, and xi, x2, • • •, £ n be n indeterminates over Z), then == bi&1 1 Gn^i G ll^i + + a a i122x#22 + + •* •' •' ++ cL CLlnXn lnxn a2iXi ++ «22^2 a22x2 H + • •• +h «o,22nnx^nn == bh 2 I «21^1 ,- ~x (1.8) o-miXi am2x2 H+ • hi ^m/i^n Q>mnXn == ^t m >J flml^l ++ «m2^2 is called a system of m linear equations in n indeterminates To use matrix notation, let A = (ay) l
A = («*j)l
( x =

Xl

x2

)

X2

?

and 6 =

b2 ?

{ bm )

30

Chapter 1. Linear Algebra over Division Rings

then (1.8) can be written in the following compact form Ax = b.

(1.9)

To find a solution of (1.8) is to find n elements ci, C2, • • •, cn in D such that when they are substituted into the m equations in (1.8), m identities will be obtained; in other words, let (

Cl

\

c2 \Cn

)

if Ac = 6, then we say that ci, C2, • • •, cn is a solution of (1.8) (or (1.9)) and, alternately, we say that the column vector c is a solution vector of (1.8) (or (1.9)). When &i = b2 = • • • = bm = 0, (1.8) (or (1.9)) is also called a system of linear homogeneous equations; otherwise, it is called a system of linear non-homogeneous equations. At first we discuss the solution of a system of linear homogeneous equations. The m-dimensional column vector with all components being 0 is denoted by

f°\

°

= :

'

then a system of m linear homogeneous equations in n indeterminates can be written as Ax = 0,

(1.10)

where A is an m x n matrix over D, x is an n-dimensional column vector whose components are the n indeterminates #i, a?2, * * •, xn, a n d 0 is the m-dimensional zero column vector.

1.4. Systems of Linear

Equations

31

Proposition 1.28: Let D be a division ring and A = (atj)i
PAQ =■

(

0

:

)

■

Let c be a solution vector of (1.10), then Q~xc is a solution vector of PAQx = 0, and conversely. Assume that the solution space of (1.10) is of dimension s. Let R be a matrix representation of the solution space of (1.10), then R is an n x s matrix of (column) rank s. Clearly Q~XR is a matrix representation of the solution space of PAQx = 0. By Proposition 1.14, Q~*R is of column rank 5, hence the solution space of PAQx = 0 is also of dimension s. Now the system of linear homogeneous equations PAQx = 0 is xi x1 == 00 " ] x22 = 0 ►

.

>.

xr = 0. 0. J,

Chapter 1. Linear Algebra over Division Rings

32 Its solution space is

it

0 \ 0 0

xr+i,xr+2,- • - ,xn G D > ,

xr+i Xr+2

V

X

n

)

which is clearly of dimension n — r. Therefore the solution space of (1.10) is also of dimension n — r. □ Proposition 1.29: Let V be a subspace of the row vector space Define VL = {*u|*u G *D(n) and i; *u = 0 for all 1

t

veV},

n

then y is a subspace of D^ \ Furthermore, if dim V = r, then dim V 1 = n — r. Similarly, for any s-dimensional subspace U of the column vector space *£><*>, define UL = {v\v G £ ( n ) and v'ti = 0 for all *u G I / } , then J7 1 is an (n — s)-dimensional subspace of D^n\ For any subspace V of DM we have (VL)L = V and any subspace U of © W we have (UL)L = U. Proof: The proof that VL is a subspace of *Z?M is easy and, hence, is omitted. Let us prove the second statement of the proposition. Let V be a subspace of D^ and assume that dimV = r. Let vx, v 2 , • • •, vr be a basis of V. Then

^2

V ^r / is an r x n matrix of row rank r. It is easy to verify that lu G V 1 if and

1.4. Systems of Linear

Equations

33

only if tu is a solution vector of the system of linear homogeneous equations (

V l

\ I *1 \ X2

V2

\ Vr I \ Xn

= 0. )

By Proposition 1.28 the solution space of the above system of equations is of dimension n — r. Hence dim VL = n — r. It is clear that V C (VL)L. Assume that dimV = r, by what we have 1 just proved dim V = n — r and dim(y" L ) J_ = n — (n — r) = r. Thus by Proposition 1.6 {VL)L = V. □ Let V be a subspace of the row vector space D^ (or the column vector space *D(n)), then the subspace VL defined in Proposition 1.29 is called the dual subspace of V. Next we discuss the solution of a system of linear non-homogeneous equa­ tions. Proposition 1.30: Let D be a division ring, A = (aij)i<,< mi i<j< n be an m x n matrix over D, and

\ bm ) be a column vector of tD^m\

Denote by V the solution space of the system

of linear homogeneous equation (1.10) Ax = 0.

If d =

d2 \dn

)

is a solution of the system of linear non-homogeneous equations (1.9) Ax = 6,

Chapter 1. Linear Algebra over Division Rings

34 then

d+V

= {d + v\ve

V}

is the set of all solution of (1.9). Proof: The proposition can be proved by direct verification and is omitted. D

Let dij (i = 1,2, • • • , m ; j = 1,2, • • • , n ) , 6t- (i = 1,2, — - ,m) G D. system of linear equations #11#1 +
a2\Xi + a22x2 H

= W

h a>2nXn = b2

O m l ^ l + flm2^2 H

The

\

I

h Omn^n = &m J

is said to be independent if the vectors ( a n , a i 2 , • * * , flln)? («21, ^22> * * * » ^ 2 n ) , * * * , (flml, «m2? * ' ' > Gmn)

are linearly independent, and it is said to be consistent if it has a solution. Proposition 1.31: Let D be a division ring, A = (fltj)i
b=

/ 61 \ b2 \bm

G tD^m\

J

Let B = (A b). Then the system of linear non-homogeneous equations (1.9) Ax = b is consistent if and only if rank A = r a n k S . Proof: Assume that rank A=ranki?, then the column vector b is a linear combination of the n columns of A, say /

bi

\

b2

\bm J

/ <*21

di +

a

12

\

«22

V am2 )

d2 + • • - +

G2n

dn.

1.5. Hermitian, Symmetric, and Alternate Matrices Then

35

f dl \ d2 \dn

)

is a solution of (1.9). Conversely, assume that (1.9) has a solution

/ dM 2

\dn I Then b is a linear combination of the n columns of A with coefficients d\, d>2, * • •, dn. Hence rank A = r a n k S . D

1.5

Hermitian, Symmetric, and Alternate Matrices

Let D be a division ring and — : a —> a

for all a £ D

be an anti-automorphism of D. — is called an involution of D if a=a

for all a G D.

Let ^

=

(atj)l
be an m x n matrix over D. Define A = (aij)i
A n n . x n matrix H over D is called a hermitian matrix if i

~H = H.

Chapter 1. Linear Algebra over Division Rings

36

Two n x n hermitian matrices Hi and H
t

PH2P.

We would like to reduce hermitian matrices to simpler forms under cogredience transformations. Define F= {ae D\a = a} and the trace map Tr :

D —> F a i—> a + a.

Sometimes we assume that Assumption 1° JF is a subfield of D and contained in the center Z of D. Assumption 2° The map Tr is surjective, i.e., any a G F can be expressed as a = a + a for some a E D. Assume that Assumption 1° holds. Then we call F the fixed field of the involution — of D. It is clear that if 1 = x + x for some x £ D then Tr is surjective, and that if D is of characteristic not two, then any a G F can be written a s a = | + | , and hence Tr is surjective. We list some examples. Example 1.1: The complex conjugate of the complex field C -:

C—»C a + P%\—> a - fii (a, (3 G R)

is an involution of C whose fixed field is R and the trace map Tr is surjective. Example 1.2: The finite field ¥q2, where q is a prime power, has an invo­ lution defined by -

:

F g 2 —-> F 9 2 a i—y

a9,

whose fixed field is ¥q. It is known (cf. Theorem 3.31 of Wan 1992b) that Tr is surjective, even if Fg2 is of characteristic 2. □

1.5. Hermitian, Symmetric, and Alternate

Matrices

37

Example 1.3: Let H = {a + pi + 1j + 8k\a,f3,~f,8e

R}

be the division ring of real quaternions whose multiplication is defined by ,•>=;» = *' = - ! , ij = —ji = &, jk = ~kj = z, ki = —ik = j . H has an involution — defined by a + Pi + jj + 8k = a — fli — jj — 8k. The fixed elements of this involution form the real field R. Since H is of characteristic not two, Tr is surjective. □ Example 1.4: Let F be a field and take the identity map of F to be the involution. Then the fixed field is F itself. If F is of characteristic not two, then Tr is surjective. But if F is of characteristic two, Tr is not surjective. In this case the hermitian matrices are the usual symmetric matrices. □ Proposition 1.32: Let D be a division ring with an involution such that the Assumptions 1° and 2° hold. Let H be an n x n hermitian matrix over D. Then H is cogredient to a diagonal matrix (ai

\

I

I'

\

o
where ~a[ = a;• ^ 0 for i = 1,2, • • •, r and r is the rank of H. Proof: Let H = (/&;*:)i 2 and that the proposition holds for all (n — 1) x (n — 1) hermitian matrices. If H = 0, our proposition holds automatically. If H =fi 0, we can assume that fin ^ 0. In fact, if An = 0, but there is a nonzero diagonal element ha with i > 1, then interchanging the first and the z-th

38

Chapter 1. Linear Algebra over Division Rings

rows of H and the first and the i-th columns of H simultaneously, we obtain an hermitian matrix which is cogredient to H and whose element at (1,1) position is not equal to zero. If all the diagonal elements of H are zeros, since H ^ 0, there is a nonzero non-diagonal element of H. We can assume that hi2 7^ 0. For if h{j ^ 0 with i < j , interchanging the first and the z-th rows and the first and the i-th columns of H simultaneously and then interchanging the second and the j-th rows and second and the j-th columns of the matrix thus obtained simultaneously, we obtain an hermitian matrix which is cogredient to H and whose element at (1,2) position is nonzero. Therefore we can assume that /

0

\

t

h12

H\2

\ H22 I

where h\2 =fi 0, H\2 is a 2 x (n — 2) matrix and *#22 = #22- Let

P=[

X 1

(n 2)

V

),

/ - /

then the element at (1,1) position oilPHP is h12\ + h12\. Since Assumption 1° holds, F is a field and 1 G F. Since Tr is surjective (Assumption 2°), there is an element x 6 D such that 1 = x + ~x> Since h\2 ^ 0, we can find an element A £ D such that x = hi2\. Then the element at (1, 1) position of tPHP is 1. Therefore we can always assume that hu ^ 0. Write H in the block form rj __ [

^11

y tH\2

#12 \

H22 ) '

where hu — hu, H12 is a 1 x (n — 1) matrix, and ^ 2 2 = #22- Let (1 y - y

-K?H12\ j(n-i) j1

then

tQHQ^y £

H22_h-itW2Hi2J,

1.5. Hermitian, Symmetric,

and Alternate

Matrices

39

where H22 ~ h^ tH\2H\2 is an (n — 1) x (n — 1) hermitian matrix of rank r — 1. By induction hypothesis H22 — h\\ tH\2H\2 is cogredient to a diagonal matrix, so is H. □ We remarked in Example 1.4 that if we regard the identity map of a field F to be the involution, then symmetric matrices over F can be regarded as hermitian matrices and that if F is of characteristic not two, then both Assumptions 1° and 2° are satisfied. Therefore we have Corollary 1.33: Let F be a field of characteristic not two and S be an n x n symmetric matrix over F. Then S is cogredient to a diagonal matrix

(

I \

ttl

\

I ' 0( n " r ) )

where at- ^ 0 for i = 1,2, • • •, r and r is the rank of S.

□

Let us come to the study of alternate matrices. From now on we assume that D = F is a field of any characteristic. Let K = {kij)i
40

Chapter 1. Linear Algebra over Division Rings

of rank 2v(< n), then K is cogredient to /

0 -1

\

1 0 0 -1

1 0 0 -1

1 0

/

\

where the number of diagonal 2 x 2 blocks is v, and is also cogredient to 0 _/W

/<") 0 Q(n-2i/)

Proof: We apply induction on n. When n = 1, K = (0) and our proposition holds automatically. When n = 2, K has the form K =

0 -ku

k12 0

If fc12 = 0, our proposition is true; if k^ ^ 0, K is cogredient to t/

1

0

o Ki

K

1

0

o

l

o Kl

-1

0

Now suppose that n > 2 and that our proposition holds for m < n. Let

K =

0 -fcl2 -&13 \ — hn

fci2 fcl3 0 fc23 -^23 0 —k2n —k'3n

\ hn hn 0

If K = 0, then our proposition is true. Suppose that K ^ 0, then there is a hj ^ 0, i < j . Interchanging the first row and the i-th row of K and

1.5. Hermitian, Symmetric,

and Alternate

Matrices

41

the first column and the z-th column of K simultaneously, we obtain an alternate matrix K\ which is cogredient to K and whose.element at (1, j ) position is non-zero. Then interchanging the second row and the j - t h row of K\ and the second column and the j - t h column of Ki simultaneously, we obtain an alternate matrix K2 which is cogredient to K\ and hence also to K and whose element at (1,2) position is nonzero. Thus we can assume that fci2 7^ 0. Then K is cogredient to \

V

1' 1

\

1

^12

\ &12

j(n-2)

t

J(""2) J

)*(

whose element at (1,2) position is 1. Hence we can assume that ki2 = 1. Write K in the block form

* ■ (

V

( - - )

B

K)> 0 }

-*B

where B is a 2 x (n — 2) matrix and KQ is an (n — 2) x (n — 2) alternate matrix. Then K is cogredient to

u ui: \

/

)

s

) (

) \

( - - )

B

K0 ) )

-*B

w

0 -1

1 0

(■ V

J

( - :

i

»

s

)

)

5

where K\ is an (n — 2) x (n — 2) alternate matrix. Therefore our proposition follows from induction hypothesis. □ Finally, we prove the following proposition. Proposition 1.35: Let F be a field of characteristic two, n be an integer > 1, and S be an n x n nonalternate symmetric matrix over F. Then S is cogredient to a diagonal matrix. Proof: Let

sS =

ik)l
Chapter 1. Linear Algebra over Division Rings

42

where 5t& = Ski (1 < i,k < n). We apply induction on n to prove our proposition. When n = 1, it trivially holds. Now let n > 1 and assume that it holds for any (n — 1) x (n — 1) nonalternate symmetric matrix. Since S is nonalterate, there is a nonzero diagonal element of 5, say s u ^ 0. Interchanging the first row and the i-th row of S and the first column and the i-th column of S simultaneously, we obtain a symmetric matrix which is cogredient to S and whose element at (1, 1) position is nonzero. Therefore we can assume that s u ^ 0. Write

5 = | ( 311

u

V

522 /

where U u — and5*22 S22= =(St*)2 Then S is cogredient = (512, (Si2,"• •• •, S, l5i n )n )and

r

( l1 5SuU n w \ \ /( 5n Sll V I' jy'u \ ) '«

{

- 1 uu \ \ (( 11 sjs^u ! */ \11_ _/( s nsn 00 \ 1I S22)\ ) \ )) - \Q 822 + 3 ^ ^ ') ' S 2 + « u *WU ) S22 { 0 2

"l

where S22 + Su" tuu is an (n — 1) x (n — 1) symmetric matrix. Let Si = 5*22 + S\i tuu. If S'I = 0, then S is cogredient to the diagonal matrix [5n,0,0, ••• ,0]. If Si is nonalternate, then by induction hypothesis it is cogredient to a diagonal matrix and, hence, is S. If Si is a nonzero alternate matrix, then by Proposition 1.34, it is cogredient to an alternate matrix of the following form

\\

(0 / 0 11 1 00 0 1 1 00

I

I 5' 0 1 1 00

\V

2 1 Q(n-2u-\)

o^- "- ))

where the number of diagonal 2 x 2 blocks is assumed to be v and, hence,

1.5. Hermitian, Symmetric,

and Alternate

Matrices

43

S is cogredient to

/1

\

0 1 1 0 0 1 1 0 0 1 1 0 Q(n-2./-l) 1

\

We have

y 1 1 0 \ (l 0 10 \i 0 1 V

0 1 1 0

\

(1 1 o \ (l 0 10 = ll 0 1)

1 1 1 0

\

.

It follows that S is cogredient to

(I

{

)

S2)'

where

(1

1 1 0

\

0 1 1 0

s2 =

0 1 1 0

I

Q(TI-2I/-1) , Q(n-2>/-l)

S2 is an (n — 1) x (n — 1) nonalternate symmetric matrix. By induction hypothesis, S2 is cogredient to a diagonal matrix. Hence S is also cogredient to a diagonal matrix. □ Corollary 1.36: Let Wq be a finite field with q elements, where q is a power of 2, n be an integer > 1, and S be an n x n nonalternate symmetric matrix over ¥q. Then S is cogredient to a diagonal matrix whose diagonal elements are either 1 or 0.

Chapter 1. Linear Algebra over Division Rings

44

Proof: By Proposition 1.35, S is cogredient to a diagonal matrix, say [ai,a , ar ,r 0, 0, 0, 0, ,••- -•-jO), ,0], [ai,«2,2 , • ' •• •, a N

v^-

n—r n—r

where «i, a l][ a i>2,--«2, • -,a • •,r,0,-«r, 0, • •,o][&r\& • •, O ] ^ 1 ,2&J • • •, b~x, 1,•,1] • • •, 1] I&iS^V? K ! ! » • " '■,l][ai,a

^3-

= fl, [>s—M1, , - ,, 11,0, , o , ••;,()]. r

n—r

a

1.6

Comments

In order to be as self-contained as possible the book starts with a chapter on linear algebra over division rings. The material of this chapter is rather standard and can be found in many books. We follow mainly Hua and Wan 1963 and Wan 1992b.

Chapter 2 Affine Geometry and Projective Geometry 2.1

Affine Spaces and Affine Groups

Let D^ be the n-dimensional row vector space over D. The vectors in D^n\ both the non-zero vectors and the zero vector, will now be called points; the cosets of D^ relative to any 1-dimensional subspace, lines; the cosets of Z)(n) relative to any 2-dimensional subspaces, planes; and the cosets of D^ relative to any (n — l)-dimensional subspace, hyperplanes. More generally, the cosets of D^ relative to any r-dimensional subspace (0 < r < n) will now be called affine r-flats, or simply r-flats. In particular, 0-flats are points, 1-flats are lines, 2-flats are planes, and (n — l)-flats are hyperplanes. An rflat is said to be incident with an 6-flat, if the r-flat contains or is contained in the s-flat. Then the point set D^ with the affine r-flats (0 < r < n) and the incidence relation among them defined above is called the n-dimensional left affine space over D and denoted by AGl(n,D). Moreover, the point (#i, a?2, • • •, xn) of AGl(n, D) is said to have coordinates xi, #2, • * •, #n- The dimension of an r-flat is defined to be r. Proposition 2.1: A hyperplane of AGl(n,D) can be regarded as the solu­ tion set of a linear non-homogeneous equation x\a\ + x2a2 + • • • + xnan = 6, 45

(2.1)

46

Chapter 2. AfEne Geometry and Projective

Geometry

where ai, a 2 , • • •, a n , 6 G D and not all ai, a 2 , • • •, a n are zeros, and conversely. Proof: Let U + v be a hyperplane, where U is an (n — l)-dimensional subspace of the n-dimensional row vector space D^ and v a vector of D^. Choose a matrix representation of [/, call it U again, then U is an ( n - l ) x n matrix of rank n — 1. By Proposition 1.28 we know that the solutions of the system of linear homogeneous equations

U

2/2

= 0

V Vn ) form 1-dimensional subspace of the n-dimensional column space tD^n\ Let *(ai,a2, • • • , a n ) be a non-zero vector in the solution space, then by Propo­ sition 1.29 the subspace U is the solution space of the linear homogeneous equation xiai + #2^2 + • • • + xnan = 0. Let v = (vi, v2l • • •, vn) and put b = vxdx + v2a2 H

(- vnan.

Then the hyperplane U + v is the set of solutions of linear non-homogeneous equation (2.1) xiai + x2a2 + • • • + xnan = b. Conversely, let (2.1) xidx + x2a2 H

h xnan = b

be a linear non-homogeneous equation. Then the set of solutions of the corresponding linear homogeneous equation zi^i + x2a2 H

h xnan = 0

is an (n — l)-dimensional vector subspace U of the n-dimensional row vector space D^n\ Let (vi, v2, • • •, vn) be a particular solution of (2.1), then the set of solutions of (2.1) is the hyperplane U + v. □

2.1. Afiine Spaces and AfEne Groups

47

The linear non-homogeneous equation (2.1) whose solutions are all the points of a hyperplane H of AGl(n, D), is called the equation of the hyperplane H. Recall that a system of r linear equations Z i a n + Z2a21 H

+ xnO>nl = &1 1

Ziai2 + x2a22 H

h xnan2 = b2 I

xidir + x2a2r H

h xnanr = br )

is said to be independent, if the column vectors ( a n , a 2 i, •••, «ni), («i2, ^22, •••, o,n2), •••, (ai r , a 2r , •••, a n r) are linearly independent, and is said to be consistent, if the system has a common solution. More generally, we have Proposition 2.2: Any r-flat (0 < r < n) in AGl(n,D) can be regarded as the set of solutions of a system of n — r independent and consistent linear non-homogeneous equations, and conversely. □ The proof of Proposition 2.2 follows also from the theory of linear equations in the same way as Proposition 2.1 and will be left as an exercise. Proposition 2.3: In

AGl(n,D),

(i) Any two distinct points are joined by exactly one line. (ii) Any three non-collinear points lie on exactly one plane. (iii) Any r + 1 points, not lying on any (r — l)-flat, lie on exactly one r-flat (0 < r < n). Proof: (i) Let x = (zi, x2, • • •, xn) and y = (i/i, y2, • • •, yn) be two distinct points in AGl(n, D). Regarding x — y = (zi — y1? x2 — y2, • • •, xn — yn) as a non-zero vector of D^n\ it spans a 1-dimensional subspace < x — y >. Then < x — y > + 2 / i s a line of AGl(n, D) which contains both x and y. This proves the existence of a line passing through x and y. Now, let U+u be a line in AGl(n, D), where U is a 1-dimensional subspace of Z)(n), and u £ D^. Assume that x, y £ U + u. Then x = u\ + w, y — u2 + u,

Chapter 2. AfEne Geometry and Projective

48

Geometry

where Ui, u2 G U. Hence U + u = U + y and x — y = ui — u2£U. Therefore U = < x — y > and U + u = < x — y > + y. This proves the uniqueness of the line passing through x and y. Since (ii) is a special case of (iii), it is enough to prove (iii). (iii) Let x^ = (x^, x2\ • • •, a^)? i = 1,2, • • •, r + 1, be r + 1 points in AGl(n,D) and assume that they do not lie on any (r — l)-flat. Let

„w = Xn-Xir+D = (xfp-x£+1\x$>-x%+1), • • •, 4' } -4 r + 1 ) ), t = i, 2, ■ • •, r, and V be the subspace spanned by u ' 1 ' , u ' 2 l , ' , , , u ' r ' . Clearly d i m V < r and V + x^r+1^ is an affine flat containing x^\x^2\ • • • ,x^r\x^r+1\ Since x^\ x&\ - • •, x^r+1^ do not lie on any (r - l)-flat, dim V = r and V + z ( r + 1 ) is an r-flat. Now let U + u be an r-flat containing x^\x^2\ • • • ,x^r+1\ where U is an r-dimensional subspace and u G D^n\ Then x^ = u ^ + u, where t i ^ G [/, * = l , 2 , - - - , r + l. Thus 17 + u = 17 + a? and sW - a? = t*W r 1 Hence U contains the subspace V spanned by w ( + ) £ C7, z = 1,2, "-,r. a.(i) _ x(r+i)jXP) - a?( r+1 ),..-,a?W - x^r+1\ Since dim 17 = dimV = r, we have U = V and U + u = U + x + yy where < x — y > is the 1-dimensional subspace spanned by the non-zero vector x — y. Hence the line passing through x and y is the point set {X(xi - yi>x2 - y2y- - ,xn - yn) + (yuy2,-

- ,yn)\X G £ } ,

or {X(xux2i--,xn)

+ (l - X)(y1,y2,'-',yn)\X

G D},

or {X(xux2,-

- ,xn) + iA(yi,y2,- •' ,yn)\X,/i

G D,X + p = 1}.

2.1.

Affine Spaces and AfEne Groups

49

These are called the parametric representations of the line passing through x and y. Similarly, the plane passing through three non-collinear points x = (xux2,-',xn), y = (2/1,2/2,-'-,2/n), and z = (zuz2, • • • ,zn) has the parametric representation {Xx + uy + vz\X, / i , i / € f l , A + /i + i/ = l } . More generally, the r-flat containing r + 1 points x^\ x^2\ • •, x^r+1\ do not lie on any (r — l)-flat, has the parametric representation

which

{A1x<1>+A2s(2)+- • -+A r+1 x< r+1 >|Ai, A2, • • •, A r+1 G D, X1+X2+- • -+A r + 1 = 1}. We call two flats U + u and V + v, where U and V are subpaces of D^ u,v e £>(n), para//c/, if C/ C V or V C U.

and

Proposition 2.4: (i) If n > 3, there exist two skew lines in AGl(n,D), which are neither parallel nor intersecting. (ii) Through any point of AGl(n,D) parallel to a given r-flat.

i.e., two lines

there passes exactly one r-flat

(iii) Two r-flats both parallel to another r-flat are parallel to each other. Proof: (i) Let V{ = < et- > be the 1-dimensional subspace spanned by e« (1 < i < n). Then V\ and V2 + (1,1, • • •, 1) are two skew lines. (ii) Let a: be a point of AGl(n, D) and U + u be an r-flat. Clearly U + x is an r-flat passing through x and parallel to U + u. Furthermore, let V + v be an r-flat passing through x and parallel to U+u. Since x G V+v,V+v = V-\-x. Since U + u and V + v are parallel, (7 C V or V C U, but both (7 and V are of dimension r, we have necessarily U = V. Hence V + v = V + x = U + x. (iii) Let U\ +ui and U2 + u2 be two r-flats and assume that both of them are parallel to another r-flat V + v. Then dim/7 t = dim V = r, which together with the parallelism of Ui + U{ and V + v implies Ui = V(i — 1,2). Hence Ui = U2. Therefore Ui + ui and U2 + u2 are parallel. □ Let U + u be an r-flat and V + v be an s-flat, i.e., U and V are subspaces of dimensions r and 3, respectively, and u,v G D^n^. The set of points

50

Chapter 2. Affine Geometry and Projective

Geometry

belonging to both U + u and V + v is called the intersection of U + u and V + u, which is denoted by (U + u)C\ (V + v) and will be shown below to be a flat, if it is non-empty. The minimum flat containing both U + u and V + v is called the join of U + u and V + v, which is denoted by (U + u) U (V + v). Denote the dimension of a flat U + u by dim (U + u). Proposition 2.5: Let U + u and V + v be flats. Suppose that (U + u) D (V + v) 7^ (j>. Then (U + u)f)(V + v) is also a flat and we have the dimension formula dim(U + u) + dim(V + v) = dim((U + u)n(V

+ v)) + dim((U + u)U(V + v)).

Proof: Let x G (U + u) n (V + v). Then U + u = U + x,V + v = V + x. Consequently

(u + u) n (v + v) = (u + x) n(v + x) = (unv) + x, which is clearly a flat. Denote the join oiU + u and V + v by W + w, where W is a subspace of D^ <mdw e D ( n ) . Let x e (U + u)n(V + v) as above. From x G U + u C W + w we deduce U + u = U + x and W + w = W + x. Similarly, V + v — V + x. Then from U + x C W + x and V + x C W + x we deduce U C W and V C W, respectively. Therefore U + V C W and ([/ + V) + x C VK + x. On the other hand, U + x C (U + V) + x, V + x C (U + V) + x; but by definition W + x is the minimum flat which contains both U + x and V + x, hence W + x C (U + V) + x. Consequently (U + V) + x = W + x = (U + u)l)(V + v). Then by Proposition 1.8, i.e., the dimension formula of subspaces of D^n\ we have dim(U + u) + dim( V + v) = dimC/ + d i m y = dim(U DV) + dim(U + V) = dim((U r\V) + x) + dim((U + V) + x) = dim((U + u) 0 (V + v)) + dim(([/ + u) U (V + v)). D

2.1. Affine Spaces and Affine Groups

51

Now we shall introduce the affine group. We call the transformation of AG\n,D) to itself

AG\n,D)-+AGl{n,D) (xi,a? 2 ,''' ,xn) '—► (xux2,

• •,x n )T + v,

^ '

)

where T £ GLn(D) and v = (ai,a 2 , * * • , a n ) € D ^ , an afl?ne transfor­ mation of the n-dimensional left affine space AG'(n, J5). Clearly, an affine transformation is a bijective map from AGl(n,D) to itself and the set of affine transformations of AG\n, D) forms a group with respect to the com­ position of maps, which is called the affine group of the n-dimensional left affine space AGl(n, D) over D and denoted by AGLn(D). If T = (tij)i
► (Zi*H H

h S n * n l + a i , a?i*i2 H

h Zn£n2 + «2, ' ' * ,

^ l ^ l n i * * * T %rJ"nn i ^ n j *

Two geometric figures in affine space are said to be affinely equivalent, if one of them can be carried into the other by an affine transformation. According to Felix Klein's Erlangen Program (1872) (cf. Klein 1921), the n-dimensional affine geometry is the study of properties of geometric figures in AGl(n, D) which are invariant under the n-dimensional affine group. Let us" now study the transitivity properties of the affine group on the flats AGl(n,D).

AGLn(D)

Proposition 2.6: (i) The affine group AGLn(D) carries r-flats in AGl(n,D) into r-flats and preserves parallelism. Moreover, it is transitive on the set of r-flats in AGl(n, D). In particular, it is transitive on the set of points of AGl(n, D). (ii) AGLn(D)

is doubly transitive on the set of points of AGl(n, D).

(iii) For n > 2, AGLn(D)

acts transitively on the set of triples of non-

collinear points. (iv) For 0 < r < n, AGLn(D) acts transitively on the set of subsets consisting of r + 1 points which do not lie on any (r — l)-flat.

52

Chapter 2. Affine Geometry and Projective

Geometry

Proof: (i) Let U+u be an r-flat in AGl(n, Z)), that is, U is an r-dimensional subspace of D^ and u G D^n\ Under the affine transformation (2.2), U + u is carried into UT + (uT + v) which is also an r-flat. Let U\ + u\ be an r-flat and U2 + U2 an s-flat in AG\n, D). Assume that they are parallel, and for definiteness we assume that U\ D U2. Under the affine transformation (2.2), U\ + U\ and U2 + u2 are carried into U\T + (uiT + v) and U2T + (u2T + v), respectively. From Ux D U2 we deduce UXT D U2T, hence UiT + {urT + v) and U2T + (u2T + v) are parallel. Now let f/i + u\ and U2 + u2 be two r-flats in AGl(n,D). By Proposition 1.24, there is an element T G GLn(D) such that UiT = U2. Then the affine transformation (zi, x2l • • •, xn) 1—y (xu x2, • • •, xn)T + (u2 -

uiT)

carries U\ + u\ into U2 + U2. This proves the transitivity of the affine group on the set of r-flats. (ii) To prove the double transitivity of the affine group on the set of points of AGl(n, D), it is sufficient to prove that any ordered pair of points Vi and V2 in AGl(n, D) can be carried under the affine group to a particular ordered pair of points, say (0,0, • • •, 0) and e\. Clearly the affine transformation (zi, x 2 , * * *, xn) 1—► (xi, x 2 , ' *', ^n) - vi

(2.3)

carries Vi and V2 into (0,0,- • • ,0) and V2 — Vi, respectively. By Corollary 1.25 there is an element T G GLn(D) such that (v2 — v\)T — ei, then the product of affine transformation (2.3) and (xi, x 2 , • • •, xn) 1—> (xi, x 2 , • • •, x n )T carries Vi and v2 into (0,0, • • • ,0) and ei, respectively. (iii) is a special case of (iv), so it is enough to prove (iv). (iv) It is easy to verify that an affine transformation carries a set of r + 1 points not lying on any (r — l)-flat into a set of r + 1 points not lying on any (r — l)-flat. We only need to prove the transitivity part of (iv).

2.1. Affine Spaces and Affine Groups

53

By Proposition 2.3(iii) any r + 1 points not lying on any (r — l)-flat lie on exactly one r-flat. By (i) AGLn(D) acts transitively on the set of r-flats in AGl(n,D). Therefore to prove (iv) it is enough to show that for any two sets of r + 1 points not lying on any (r — l)-flat but lying on the same r-flat there is an affine transformation which carries one of the two sets into the other. Clearly, 0 = (0,0, • • •, 0), ei, e 2 , • • •, er are r + 1 points not lying on an (r — l)-flat. They lie on the r-flat U = {(xi, x 2 , • • •, x r , 0,0, • • • , 0 ) 1 ^ G £>}. It is enough to show that for any r + 1 points v 0 ,t;i, v2, • • • ,v r not lying on any (r — l)-flat but lying on [/, there is an affine transformation which carries v0, vi, v2, • • •, vr into 0, ei, e 2 , • • •, e r , respectively. Clearly the affine transformation (xi, x 2 , • • •, x n ) i

> (xi, x 2 , • • •, x n )vo

carries vo into 0 and V{ into vt- — Vo for i = 1,2, - - ^ r . By hypothesis, ^o,^i5^2? • • • ,u r do not lie on any (r —l)-flat, so do 0, v\— v0, v2 — t>o, * • *, vn — VQ. But Uj — v0 G C/ for i = 1,2, • • •, r. Hence t>i — v0, v2 — ^o, • • •, v r — v0 are r linearly independent vectors of U. Then /

Vi

-V0

\

v2 — VQ

G GLn{D)

T = Vr -

Q(n-r,r)

V0

j{n-r)

and (vi - v0)T * = et- i = 1,2, • • •, r. The affine transformation (xi, x 2 , • • •, x n ) i—> (xi, x 2 , • • •, x n ) T - 1 leaves 0 invariant and carries V{ — v0 into e» for z = 1,2, • • •, r. From Proposition 2.6(i) it follows that the property of a geometric figure be­ ing an r-flat is invariant under the affine group, any two r-flats are affinely

□

54

Chapter 2. AfRne Geometry and Projective

Geometry

equivalent, and the set of r-flats is an orbit under the afEne group. From Proposition 2.6(iv) it follows that the property of a geometric figure consist­ ing r + 1 points not lying on any (r — l)-flat is invariant under the affine group and any two such figures are afflnely equivalent. Similarly, starting from the n-dimensional right vector space '£><"> formed by the n-dimensional column vectors over D we can define the n- dimensional right affine space over D, whose points are the n-dimensional column vectors over D and whose r-flats are the cosets of *D(n) relative to any r-dimensional subspace of fD^ (0 < r < n). The n-dimensional right affine space over D will be denoted by AGr(n, D). Parallel to the above results of AGl(n, D) we have the corresponding results of AGr(n, Z)), but the details will be omitted. When D = F is a field, we have identified the n-dimensional right vector space lF^ with the n-dimensional left vector space F^ in Section 1.1 and write lF^ also as F^n\ We can also identify the n-dimensional right affine space over F , A£? r (n,F), with the n-dimensional left affine space over F , AGl(n,F), and write AG{n,F) for both AGr(n,F) and AGl{n,F).

2.2

Fundamental Theorem of the AfRne Geometry

By Proposition 2.6 we know that any affine transformation of the n-dimen­ sional left affine space AGl(n,D) over a division ring D carries r-flats into r-flats (r = 0, l , - - - , n ) . In particular, it carries lines into lines, planes into planes, etc. Now we would like to characterize the affine transforma­ tions with as few invariants as possible. Actually, in most of the cases the affine transformations of AGl(n,D) can be characterized as bijective maps of AGl(n,D) to itself, carrying lines into lines, to within automorphisms of D. More precisely, we have the following fundamental theorem of the affine geometry over any division ring. Theorem 2.7: Let D be any division ring, n be an integer > 2, and A be a bijective map of the n-dimensional left affine space AGl(n, D) to itself which carries lines into lines. When n > 3 and D = F2, assume further that

2.2. Fundamental Theorem of the Affine

Geometry

55

A carries planes into planes. Then A is of the form A(xu

x2, - ■ •, xn) = (xi, x 2 , • • •, xnfT

+ (ai, a 2 , • ■ •, a n )

(2.4)

for all (xi,x 2 , • • •, £ n ) G AGl(n,D), where <J is an automorphism of D, a ( x ! , x 2 , . - - , a ; n ) = ( 4 , 4 , • • • , < ) , T G GLn(D), and (ai,a 2 , • • • , a n ) G Z)(n). Conversely, any map of the form (2.4) from AGl(n,D) to itself is bijective and carries lines into lines, planes into planes, and r-flats into r-flats (0 < r < n). We need some lemmas. L e m m a 2.8: Let D be any division ring, n be any integer > 1, and A be a bijective map of AGl(n, D) to itself which carries lines into lines. Then A * carries lines into lines, too. If A carries also planes into planes, then so does A-1. Proof: It is enough to prove that any line of AGl(n,D) is the image of a line of AGl(n,D). Let / be any line of AGl(n,D) and P i , P 2 be two distinct points on /. Then ^4 _1 (Pi) and *4 -1 (P 2 ) are two distinct points of AGl(n,D). There is a line passing through A~1(P\) and ^4 _ 1 (P 2 ), and call it by /'. Clearly A(V) — /. Hence A'1 (I) = V is a line. The second statement can be proved in the same way as the first one. □ L e m m a 2.9: Let n > 2 and A be a bijective map of AGl(n,D) to itself which carries lines into lines. When n > 3 and D = F 2 , assume further that A carries planes into planes. Then both A and A'1 carry also r-flats into r-flats for any r with 0 < r < n. Proof: By Lemma 2.8 it is enough to prove the lemma for A. We apply induction on r. By hypothesis our lemma holds for r = 0 and 1. Now let r > 2 and assume that A carries (r — l)-flats into (r — l)-flats. We are going to prove that A carries also r-flats into r-flats. Let S = V + a be an r-flat, where V is an r-dimensional subspace of D^ and a is an ndimensional row vector of D^n\ Choose any basis v1? v2, • • •, vr of V, then V = < vi, u2, • • •, vr >. Let S' = A(S) and .4(a) = a' G 5". Define V' = {vfeDW\v'

+

a'eS'},

56

Chapter 2. Affine Geometry and Projective

Geometry

then S' = V + a1. We need to prove that V is an r-dimensional subspace. Let Vr_i = < v1,v2,-' ,vr-X > and W = < vr > . Then V = Vr-i+W, Vr-if)W — < 0 > , Vr-i + a is an (r — l)-flat contained in S, and W + a is a 1-flat (or line) contained in 5 . By induction hypothesis and the hypothesis of the lemma, A(Vr-i + a) is an (r — l)-flat contained in S' and A(W + a) is a 1-flat contained in S\ respectively. Therefore we can assume that A(Vr^

+a) = V;_, + a'

and A{ W + a) = W' + a\ where V^_x is an (r — l)-dimensional subspace of D^ and W is a 1dimensional subspace of Z?(n). Then V^x + a' C S' and W + a' C S'. By the definition of V K-i Q V

and

W C V.

We assert that

v' = vj!_1 + wi and y ; _ 1 n ^ , = < o > . From these two assertions it follows immediately that Vf is an r-dimensional subspace of D^. If K - i n ^ ^ < 0 > , then (!£_! + a') D (W7 + a') ^ {a 7 }. It follows that ( K - i + a) H (W + a) ^ {a} and then Vr_i fl W ^ < 0 >, which is a contradiction. Thus the second assertion is proved. Now let us prove the first assertion. We distinguish the following two cases. (a) D ± F 2 . Let v' be any element of V. Then v'+a' G S'. Since A(S) = S' and 5 = V + a, there is an element v EV such that A(v + a) = v' + a!. We may decompose v &s v = u + w, where u G Vr-i and w G W. Let X £ D but A ^ 0 , 1 . Then Au G V r _i. Denote the line joining the points Xu + a and v + a by /i, then /a = {xXu + (1 - x)v + a|x G £ } .

2.2. Fundamental Theorem of the Afhne

Geometry

57

Denote the line joining the points a and w + a by Z2, then /2 = {xw + a\x G D}. /1 and l2 intersect at a point which isA(A — 1 ) - 1 K ; + a. Thus l\ is the line passing through \u + a, A(A — 1) _ 1 K; + a, and v + a. Let ,4(Au + a) = u + a

and ,4(A(A - l) _ 1 u; + a) = w' + a,

where u' G V^_x and u/ G W . Then u' + a', u/ + a\ and u' + a' lies on the line which is the image of l\. Consequently, v' = \iv! + (1 — fi)wf for some fi e D. Since u' G K'-i

and

™' £ W>

we nave v

' £ K'-i + W'•

Therefore

v c y;.! + w. Conversely, for any u' G V ^ and w' G W7 let v' = uf + w'. Let A G D, but A 7^ 0,1. Then \u' G V ^ . Denote the line joining the points \u' + a' and v' + a' by l[, then l[ = {x\uf + (1 - z)}. Denote the line joining the points a' and w' + a' by /^ then 1'2 = {xw' + a'\x G D). l[ and /^ intersect at a point which is A(A — l ) _ 1 u / + a'. Thus l[ is the line passing through \u' + a', A(A — l ) - 1 u / + a', and v' + a'. Since \u' G V^_1 and A(A — l ) _ 1 u / G W , there are u G K—i and w G VK such that A(u + a) = \u + a and .4(u; + a) = A(A - 1 ) ~ V + a. By Lemma 2.8, l[ is the image of the line joining u + a and w + a under A, and t/ + a' is the image of a point on that line under A. Thus v' + af G S" and v7 G V7. Therefore V / ^ + W C V'. Assembling the two inclusions obtained in the above two paragraphs, we obtain V = K - i + W'. (b) D = ¥2. In this case, by hypothesis A carries not only lines into lines, but also planes into planes.

58

Chapter 2. Affine Geometry and Projective

Geometry

Let v' G V. Then there is an element v G V such that A(v + a) = v' + a'. we decompose v&sv = u + w, where u G Vr-i and it; G W. Let ,4(u + a) = w' + a' and A(w + a) = u/ + a', where u' G V?_i and tt/ G W. < u > + a is the line passing through u + a and a, and < u' > + a' is the line passing u' + a' and a'. By hypothesis A(< u > + a) = < u' > + a'. Similarly, A(< w > + a) = < wf > + a'. < u,w > + a is the plane containing the lines < u > + a and < w > + a, and < w', u/ > + a' is the plane containing the lines < v! > + a' and < w' > + a'. By hypothesis, A(< u,w > + a) = < u ' , u / > + a'. Since u + a = w + u; + a G < w , ^ > + a, we have v' + o! = A(v + a) G -4(< u, w > + a) = < u\ w' > + a'. Hence vf = Au' + /xiy', where A, JU G D. Thus v' G V^.j + W. Therefore V C y / . ! + W . Conversely, let u' G V/.! and u;' G W ; . Then there is a u G K - i and a w £ W such that »4(u + a) — u + a' and A(w + a) = u/ + a'By hypothesis, ^4 carries lines into lines. Thus A carries the line < u > + a through a and u + a into the line < u' > + a' through a' and u' + a', and similarly, A carries the line < w > + a into the line < w' > + a'. By hypothesis, A carries also planes into planes. Thus A carries the plane < u,w > + a containing the two distinct lines < u > + a and < w > + a into a plane containing the distinct lines < u' > + o! and < w' > + a', which must be < v!', w' > + a'. Hence S' contains the point u' + w' + a' on the plane. Therefore u' + w' G V. This proves that V^_x + W C V . Hence in this case we also have V = V^._1 + W.

D

Corollary 2.10: Let n > 2, A be a bijective map of AGl{n,D) to itself which carries lines into lines, and .4(0) = 0, where 0 = (0,0, • • • ,0). When n > 3 and D = F 2 , assume further that A carries planes into planes. Then A carries any set of linearly independent row vectors into a set of linearly independent row vectors.

2.2. Fundamental Theorem of the AfBne

Geometry

59

Proof: Let v1,v2,- " ->vr be r linearly independent row vectors in Then r < n. Let V = < vi, v2, • • •, vr >, then V is an r-flat in D^n\ By Lemma 2.9 4 ( V ) is an r-flat in AGl(n,D). Since 0 G < ui, t>2, • * *, vr > and -4(0) = 0 , 4 ( V ) is an r-dimensional subspace of D^n\ We apply in­ duction on r to show that A(vi), Afa), - • -, A(vr) are linearly indepen­ dent. If r = 1, then v\ ^ 0. hence A(v\) ^ 0 and A(v\) is linearly independent. Assume that A(vi),A(v2), • • • , 4(tV-i) are linearly indepen­ dent. Let Vr-i — < vi, U2,- • •, vr-i > and W = < vr >. Then V = K - i + W and K_i fl W = < 0 > . By the proof of Lemma 2.9, A(V) = K'-i + W' and !//_! H W = < 0 >, where V / ^ = 4 ( K _ i ) and W = A{W). Clearly, ^ M W . - M K - i ) € K'-i and 4 ( v r ) G W. There­ fore 4(i>i), -4(^2)?'' * 5 4(iV-i), 4 ( v r ) are linearly independent. □ Now let us come to the proof of Theorem 2.7. Proof of T h e o r e m 2.7: The proof of the second statement is the same as the proof of Proposition 2.6(i) and hence is omitted. Now let us prove the first statement. We prove by induction on n. Consider first the case n = 2. Let A be a bijective map of AGl(n, D) which carries lines into lines. By the bijectivity of A, A carries parallel lines into parallel lines. Clearly, (0,0), (1,0), and (0,1) are three non-collinear points. By hypothesis, .4(0,0), ,4(1,0), and .4(0,1) are three non-collinear points, too. By Proposition 2.6(iii) there is an afEne transformation which carries .4(0,0), .4(1,0), and 4 ( 0 , 1 ) into (0,0), (1,0), and (0,1), respectively. Thus after subjecting A to this affine transformation, we can assume that 4 ( 0 , 0 ) = (0,0), 4 ( 1 , 0 ) = (1,0), and 4(0,1) = (0,1). We distinguish the following two cases. (a) D = F 2 . 4G(2,F 2 ) has only four points (0,0), (0,1), (1,0), (1,1). Therefore we must have .4(1,1) = (1,1). Thus A(x,y)

= (x,y) for all (x,y) €

AG(2,¥2).

60

Chapter 2. AfRne Geometry and Projective

Geometry

Hence Theorem 2.7 is proved for the case n = 2 and D = F2. (b) D ^ F 2 . Denote the line passing through (0,0) and (1,0) by /, and the line passing through (0,0) and (0,1) by /*. Then I = {(x,0)\x

e D]

and l* =

{{09x)\xeD}.

By hypothesis A(l) — I and A(l*) = /*. We can assume that A{x,0) = (x%0). We assert that a is an automorphism of D. Clearly a is a bijection and 0* = 0, la = 1. We have to prove that (s + t)ff = s° + ta and (sty = s't*

for all s,t e D.

These two formulas can be proved by the method of intersection and join, which was shown by the following figures.

Figure 1

Figure 2

2.2. Fundamental Theorem of the Afhne Geometry

61

The proof of the first formula reads as follows. Let (a, /?) be a point not on / and not on /*. Then a / 0 and (3^0. Passing through the point (0,0) draw the line lo =

{x(a,P)\xeD}.

Clearly, l0 ^ I and / 0 fl/ = {(0,0)}. (a, f3) is a point of the line l0 and distinct from (0,0). By Proposition 2.4(ii), passing through (a, /?) there is exactly one line parallel to /. Denote this line by /' , then l' = {(x,0) +

(a,/3)\xeD}.

There is a unique line passing through (s,0) and parallel to Z0, which is ls = {x(a,(3) +

(s,0)\xeD}.

The lines ls and V intersect at a point, denote it by P , then

P = (s + a,P). Denote the line joining (t, 0) and (a, 0) by lt, then lt = {x{t,0) + = {x(t-a,-P)

+

(l-x)(a,/3)\xeD} (a,/3)\xeD}.

Denote the line passing through P and parallel to lt by / s + t , then ls+t = {x(t - a, -P) + (s + a, P)\x e D}. The point of intersection of / and l9+t is (s + *,0), which is independent of the particular choice of the point (a, /?) ^ /, /'. In the above discussion if we replace (s,0) and (*, 0) by ( s a , 0) and (ta, 0), respectively, then (s +1,0) will be replaced by (sa + ta, 0). Therefore (s + t)a = sa + ta. Now let us prove the second formula. Since D ^ F 2 , there are at least two distinct lines passing through (0,0) and ^ /,/*. Let them be l0 =

{x{a,(3)\xeD}

and l'0 =

{x(a1J1)\xeD}.

62

Chapter 2. AfRne Geometry and Projective

Geometry

Then a, /?, a i , /?i are all ^ 0 and (a, /?) and (c*i, /?i) are linearly independent. Clearly, / o n / = { ( 0 , 0 ) } . Passing through (1,0), draw the line /1 = {x(a 1 ,^ 1 ) + ( l , 0 ) | a : G i ) } , which is distinct from / and not parallel to IQ. Denote the point of intersection of l0 and /i by Pi, then P1 = ((a - / ? / ? ! - % r ' a , (a - / ^ a x ) " 1 / ? ) . There is a unique line passing through («s,0) and parallel to /i, denote it by / s , then ls = {x{au/31) + (s,0)\xeD}. Denote the point of intersection of Z0 and ls by P-i, then P2 = (s(a - / ^ a i r V ^ a - ^ r 1 ^ ) " 1 / ? ) Denote the line passing through (t,0) and Pi by lt, then h = {x(t, 0) + (1 - *)((« - flST1^)-1**, (a - / J / J r V ) " 1 / ? ) ! * € £>} = {x(< _ ( a _ ftSr'ai)-1^ - ( a - / J / J r 1 ^ ) " 1 / ? ) +((a fifr-1^)-1^ (a - ^ r ' a i ) - 1 ) / ? ) ^ € Z?}. The line passing through P2 and parallel to /t is

/* = {*(* - (a - flST1**!)-1)^ - ( a - ^ r 1 ^ ) / ? ) +(s(a - ^ r

1

^)"

1

^ *(a - ^'1a1)'1P)\x

€ £>}.

The point of intersection of / and Zs* is (s£,0), which is independent of the particular choice of the lines IQ and lf0. If in the above discussion we replace (s,0) and (t, 0) by (s% 0) and (t a ,0) respectively, then (st, 0) will be replaced by (sata,0). Therefore (st)a = s°t°. Hence we have proved that a is an automorphism of D. After subjecting A to the bijective map of AGl(2, D) to itself (x,y)\—►

(x,y)a~\

which is of the form (2.4), we can assume that •4(3,0) = (3,0) for all

xeD

2.2. Fundamental Theorem of the Afhne Geometry

63

and 4 ( 0 , 1 ) = (0,1). By hypothesis the line /* = {(0,y)|y G D) passing through (0,0) and (0,1) is carried into itself by A. Therefore A(0,y) = (0,y T ) for all y G D, and we can prove in a similar way that r is an automorphism of D. The line passing through (s,0) and parallel to /* is {(0,y) + ( * , < % € ! > } , the line passing through (0, i) and parallel to I is {(s,0) + ( 0 , t ) | * € Z > } , and their point of intersection is (s,t). A(s,t)

Thus

= (s, T ) .

The line passing through (0,0) and (s,t) is {x(s,t)\x A(x(s,t))

= (xp(s,tT))

(2.5) G D}. Therefore

for alia; G D,

where /> is a bijection of Z). That is, A{xs,xt)

= (xps,xptT)

for all x e D.

(2.6)

On the other hand, substituting xs and xt for 5 and £, respectively, in (2.5), we obtain A(xs,xt) = (xs,xTtT) for alls G D. (2.7) Comparing (2.6) and (2.7), we obtain p = T = 1. Therefore ,4(s,t) = (s,t) for all s,t e D. Hence Theorem 2.7 is proved for the case n = 2.

Chapter 2. Affine Geometry and Projective

64

Geometry

Now let n > 3 and assume that Theorem 2.7 is true for the (n — 1)dimensional left affine space over D. We are going to prove that Theorem 2.7 is also true for the n-dimensional left affine space over D. Let 4 be a bijective map of AG ! (n, D) to itself which satisfies the conditions of Theorem 2.7. After subjecting A to the map (xu x2, • • •, xn) i—> A(xu x2, - - •, xn) - .4(0), where 0 = (0,0, • • •, 0), we can assume that 4 ( 0 ) = 0. Let A(ei) = (an, ai2, • - •, a t n ),

i = 1,2, • • •, n.

By Corollary 2.10 the n row vectors ( a n , a i 2 , • • • , ^ l n ) , («21> «22? • ' ' , «2n)? ' * * ? ( ^ n l , O n 2, * • • ,

ann)

are linearly independent. Let / an

a12

• • • aln \

«21

«22

'* '

#2n

\ «nl

«n2

* * '

Ann /

j. _

-* then T £ GLn(D).

I

I>

After subjecting 4 to the affine transformation (xux2,

• • •, xn) i—> (si, s 2 , • • •, ^ n ) T _ 1 ,

we can assume that 4(0) = 0 and 4(e z ) = et- for z = 1,2, • • •, n. Let A n _i = {(xi,x2,--'>xn-i,0)\xuX2,-'-,xn-i

e D},

then 4 n _ i is an (n — l)-flat containing the points 0, ei, e 2 , • • •, e n _i. By Lemma 2.9 4 ( A n _ i ) is an (n —l)-flat containing the points 0,ei,e 2 , • • • ,e n _i. By Proposition 2.3(iii) such a flat is unique. Therefore 4 ( A n _ i ) = 4 n _ i . Hence 4 induces a bijective map of 4 n - i to itself, which satisfies the condi­ tions of Theorem 2.7. By induction hypothesis, we have A(x1,---,xn-u0)

= {x1,---,xn_1,0y

IT

o\

l

l

Q*

J+(a1,---,aB_i,0),

2.2. Fundamental Theorem of the Affine Geometry

65

where a is an automorphism of D, T\ G GLn-i(D)\ and «i, • • • , a n - i £ £)• Since .4(0) = 0, we have ai = ••• = a n _i = 0. After subjecting A to a bijective map of the form (2.4)

(xi,x ( # 1 , 2#,-2,

• • ' •?, X£n)n )

1

►

(xi,x2,- • • i * n )

'Ti (

(P - 1 "

a-1

1

we can assume that A(xux2,,"

••• ,x ,^n-i,0) , z n _i,0) _ i , 0 ) = (xi,x (xi,x n _i,0) 2,-2 ,-- '•,x

and A(e n ) =: e„. As in the proof of case n = 2, we can prove that *4(zi,:r 2 , • • - , , £ „ ) = ( ^ i , « 2 , - " - ^ n )

for all (xux2,

• • • ,x n ) € A G ^ n , / ? ) ,

but the details will be omitted.

D

Remark 2.1: For the case D = F 2 and n > 3 if we assume only that the bijective map of AC?(n,F2) to itself carries lines into lines, then we cannot deduce that the map is of the form (2.4). In fact, when D = F 2 , every line of AG(n, F 2 ) contains only two points, and therefore any permutation of the points of AG(n,F2) carries lines into lines. Without any essential difficulty, Theorem 2.7 can be generalized as follows. Theorem 2.11: Let n and n' be integer > 2, and D and D' be division rings. Let A be a bijective map from the rc-dimensional left or right affine space An = AG\n,D) or AGr(n,D) to An, = AGl{n',D') or AGr(n\D'). Assume that A carries lines into lines. When n > 3 and D = F 2 , assume further that A carries planes into planes. Then n = n' and D is isomorphic to D' or anti-isomorphic to D'. In the first case, An and Ant are either both left affine spaces or both right affine spaces. If An = AGl(n,D) and Ani = AGl(n, D'), then A is of the form A(^i,x2,- • '

• ,xn)aT

? # n ) — V 3 '!? ^ 2 ? * *

+ (ai,a 2 ,« • • j ^ n )

Chapter 2. Affine Geometry and Projective

66

Geometry

for all (xu • • •, xn) G AGl{n, D); if An = AGr{n, D) and An, = AGr{n, £>'), then A is of the form A\xi,

x2, - • •, xn) = T[t(xux2l

• • •, xn)]a + *(<*!, a 2 , • • •, a n )

for all ' ( ^ i , ^ , - • • , £ n ) € AG r (n,Z}); in the above two formulas a is an isomorphism from D to D', T G GLn(Df), and a i , a 2 , • • •, an e D'. In the second case, one of An and An/ is a left affine space and the other one is a right affine space. If An = AGl(n,D) and An> — AGT(n,D'), then A is of the form A(xu x2, • • •, xn) = TftaJi, x 2 , • • •, xn)]T + *(ai, a 2 , • • •, an) for all (xu • • •, xn) e AGl{n, Z>); if A n = A<3r(n, £>) and A n , = AG^n, D'), then ^4 is of the form At(xi,x2,

• • •, xn) = (xi, x 2 , • • •, xn)TT + (a x , a 2 , • • •, a n )

for all *(a:i, x 2 , • • •, xn) G AGr{n, D)\ in the above two formulas r is an antiisomorphism from D to D', T G GLn(D'), and ai, a 2 , • • •, an G i?'. D

2.3

Projective Spaces and Projective Groups

Let n be an integer > 1 and Z)(n+1) be the (n + l)-dimensional row vector space over D. The 1-dimensional subspaces of Z)(n+1) will now be called points, the 2-dimensional, 3-dimensional, and n-dimensional subspaces of J}(n+1) w iU be called lines, planes, and hyperplanes, respectively. More gen­ erally, the (r + l)-dimensional subspaces of Z)(n+1) will be called projective r-flats, or simply r-flats (0 < r < n) in this and next sections. Thus 0-flats, 1-flats, 2-flats, and (n — l)-flats are points, lines, planes, and hyperplanes, respectively. An r-flat is said to be incident with an s-flat, if the r-flat as a subspace contains or is contained in the s-flat as a subspace. Then the set of points, i.e., the set of 1-dimensional subspaces of D^n+1\ together with the r-flats (0 < r < n) and the incidence relation among them defined above

2.3. Projective Spaces and Projective

Groups

67

is called the n-dimensional left projective space over D and is denoted by PGl(n,D). To be more concrete, we introduce the coordinate description of PGl(n, D). Let P be a point of PGl(n,D), that is, P is a 1-dimensional subspace of Z)( n+1 ). Let (x0, xi, • • •, xn) be a non-zero vector in P , then P =

{\(x0,xU'-,xn)\\eD}.

For any A € D*, we shall call the non-zero vector (Ax0, A#i, • • •, Xxn) a system of coordinates or simply the coordinates of the point P. We also say that (Arco, A#i, • • •, Xxn) is the point P. Clearly, a system of coordinates of a point P is uniquely determined up to a non-zero constant multiple of D from the left. According to the above definition of PGl(n^D)^ the set of r-flats of PGl(n,D) and the set of (r + l)-dimensional subspaces of Z)(n+1) are in one-to-one cor­ respondence. The r-flat F corresponding to an (r + l)-dimensional subspace U can be regarded as the set of points whose coordinates are the non-zero vectors of U. After adopting this point of view, we have Proposition 2.12: A hyperplane in PGl(n,D) is the set of points whose coordinates are non-zero solutions of a linear homogeneous equation in n +1 unknowns zo^o + x\a\ + • • • + xnan = 0,

(2.8)

where ao, ai, • • •, an 6 D are not all zero, and conversely. Proof: Let H be a hyperplane of PGl(n,D), then H corresponds uniquely to an n-dimensional subspace U of Z)(n+1) such that H is the set of 1dimensional subspaces contained in U. Let v i , v 2 , - - - , v n be a basis of £/", then (Vl\

\Vn

)

is a matrix representation of the n-dimensional subspace U. Consider the

68

Chapter 2. AfEne Geometry and Projective

Geometry

system of linear homogeneous equations in n + 1 unknowns

( Xx ° \ i

U\

.

=0.

\ Xn )

Since rankf/ = n, the solutions of this system of equations form a 1dimensional subspace P of tD^n+1\ Let (a 0 , ai, • • •, a n ) be a non-zero vector of P , then the n-dimensional subspace U is the solution space of the linear homogeneous equation in n + 1 unknowns (2.8). Hence the hyperplane H is the set of points whose coordinates are non-zero solutions of (2.8). Conversely, let (2.8) be given. Then the set of solutions of (2.8) form an ndimensional subspace U of D^n+1^, and hence the hyperplane corresponding to U is the set of points whose coordinates are non-zero solutions of (2.8). D

The linear homogeneous equation (2.8), whose non-zero solutions are all the points of a hyperplane if, is called the equation of the hyperplane H. Clearly, it is uniquely determined up to a non-zero constant multiple of D from the right. In the same way we can prove also Proposition 2.13: Any r-flat (0 < r < n) of PGl(n, D) is the set of points whose coordinates are non-zero solutions of a system of n — r independent linear homogeneous equations in n + 1 unknowns, and conversely. □ We define the dimension of an r-flat F in PGl(n,D) to be r and denote d i m F = r; however the corresponding (r + l)-dimensional subspace U is of dimension r +1. Moreover, we define the empty set (f> of points in PGl(n, D) to be of dimension —1 and denote dim
2.3. Projective Spaces and Projective

Groups

69

respectively, then the corresponding subspace of JF\ fl F2 is U\ C\ U2 and the corresponding subspace of Fi U F2 is U\ + U2. From the dimension formula of Z)( n+1 ), we deduce immediately the dimension formula of PGl(n, D). Proposition 2.14: Let F1 and F2 be two flats of PGl(n,D).

Then

dimFi + dimF 2 = dim(f\ fl F2) + dim(F1 U F2). D

Proposition 2.15: In

l

PG (n,D),

(i) Any two distinct points are joined by a unique line. (ii) Any three points, which are not collinear, lie on exactly one plane. (iii) Any r + 1 points, which do not lie on any (r — l)-flat, lie on exactly one r-flat (1 < r < n). Moreover, on any r-flat (1 < r < n) there exist r +1 points, which do not lie on any (r — l)-flat. Proof: We need only to prove (iii). Let P 0 , A , * • * > Pr be r + 1 points which do not lie on any (r — l)-flat. Let (a0^\ai^\ • • • , a n ^ ) be the coordinates of Pi (0 < i < r). Denote the subspace spanned by the r + 1 vectors (a 0 (l '\ a i « , • • •, a n W), i = 0,1, • • •, r, by U. Then dim U < r + 1. Denote the flat corresponding to U by F. By hypothesis, d i m F > r, and, hence, dimU = d i m F + 1 > r - f 1. Consequently, dimf/ = r + 1 and, hence, dimF = r. Of course, F contains Po,Pi, — - ,Pr. This proves the existence part of the first assertion in (iii). Now let F' be another r-flat containing Po, P\,' • •, Pr • Then there is a subspace U' of dimension r + 1 corresponding to F'. Thus P0> Pi, •'' ? Pr and, hence, U are contained in U'. But dim U = dim U' = r + 1, hence [/ = U' and therefore F = Ff. Now let F be an r-flat and U be the corresponding subspace of dimension r+1. Let vo, ui, • • •, vr be a basis of [/. Then v0, Vi, • • •, vT are the coordinates of r + 1 points P 0 , A?" * * 5 ^r 5 which do not lie on any (r — l)-flat. □ Any r + 1 points which do not lie on any (r —l)-flat are said to be independent and any set of r + 1 independent points is simply called an independent set ofr + 1 points. From the proof of Proposition 2.15 follows also the following lemma.

Chapter 2. Affine Geometry and Projective

70

Geometry

L e m m a 2.16: Let P 0 , Pu • • •, Pr be r + 1 points in PGl(n, D) and P t have coordinates (<*$) (0 < i < r). Then P 0 , P \ , ' ' ' >Pr iorm an inde­ pendent set of r +1 points if and only if the r + 1 vectors (ao , a i \ " * ■> an°^)> (ao , a i \ '" i a l^)> '' '•> (ao\

a

i \ '" i an^)

are

linearly independent.

□

According to Proposition 2.13 a line can be regarded as the set of points whose coordinates are non-zero solutions of n — 1 independent linear homo­ geneous equations in n + 1 variables. However as in the affine case a line in PGl(n,D) has also a parametric representation. Let P and Q be two dis­ tinct points on a line / with coordinates (x 0 , xi, • • •, xn) and (yo, yi, • • •, yn), respectively. Then / = {A(zo, £i, • • •, xn) + /i(t/o5 2/i, • • •, y n )|A, fi € D and are not both zero}. More generally, let P 0 , A , • * * ,Pr be r + 1 points not lying on any (r — l)-flat and let the coordinates of P t be {x0^\xi^\ • • •, x n ^ ) (0 < z < r). Then the unique r-flat passing through P 0 , Pi, • • •, Pr is {Ao(ar0(0), • ■ •, *„<°>) + \i(x0ll),

• ■ •, «, (1 ») + • ■ • + A r (x„ ( r \ • • •, *„('>)},

where Ao, Ai, • • •, Ar € D and are not all zero. Now let us introduce the projective group. Any T £ GLn+i(D) the following way:

defines a point to point transformation of PGl{n, D) in

PG\n,D) — PG'(n,D) ( x 0 , x i , - - - , x n ) i—> (xo,Xi,---,x n )T.

^ ' '

This is well-defined. In fact, if (yo^yir " ^Vn) is another system of coor­ dinates of the point (#o,#i, • • • , x n ) , that is, there is a non-zero element XeD such that (y0,2/i, •••,»«) = A(s 0 ,3i, • • • ,ar n ), then (y 0 ,!/i, • • • ,y n )T = A(x 0 ,a;i, • • • ,xn)T. The transformation (2.9) is called a projective transfor­ mation of PGl(n, JD), and denoted by T. Denote the center of D by Z, then we have L e m m a 2.17: Two elements T and I \ of GLn+\(D) define the same pro­ jective transformation if and only if there is an A 6 Z* such that I \ = AT.

2.3. Projective Spaces and Projective

Groups

71

Proof: Suppose that there is an element \ e Z* such that ?\ = AT. Then under the projective transformation 2\, defined by 2\, we have (x 0 , xi, • • •, i n ) i—► (x 0 , xi, • • •, x„)Ti = (x 0 , Xi, • • ■, xn)\T = A(x 0 ,xi,--- ,x n )T. Hence T\ = T. Conversely, suppose that T\—T.

Let

T

i = (5u)o
and

T

= (^')o<*,j
Then for each i (0 < i < n) t{F\ and e t T are proportional, thus there is an element At G D* such that ( 5 t 0 ) 5 i l j * * * i sin)

=

A ^ t f o , 6ii, * * * , ttn)'

But for all i ^ j , (et + ej)I\ and (e; + ej)T are also proportional, it fol­ lows that A0 = Ai = • • • = An = A, say. Therefore T\ = AT. Then for any point (xo,xi, • • • , x n ) , (xo,xi, • • • , x n ) and (xo,Xi, • • • , x n ) T i T _ 1 = (xo, xi, • • •, xn)A are proportional, thus there is an element fix G D* such that (xo,xi, • • • ,x n )A = /^(xo,Xi, • • • , x n ) . Then xoA = /zxxo,xiA = [ixx\. As­ sume that xo 7^ 0, then (xoA) _1 (xiA) = (fJ>xxo)~1(fjlxxi). Thus A -1 xo _1 xiA = xo _ 1 xi. But xi can be any element of D, so A G Z*. □ In virtue of Lemma 2.17, it is natural to introduce the factor group PGLn+1(D)

=

GLn+1(D)/Zn+u

where

Zn+1={\lW\\€Z*}. We leave to the reader to prove that Zn+i is the center of GLn+\{D). The image of T G GLn+i (D) in PGLn+i (D) is also denoted by T and is a projec­ tive transformation. The group PGLn+1(D) is called the projective general linear group of degree n + 1 over D. Two geometric figures in the projective space are said to be projectively equivalent, if one of them can be carried into the other by a projective transformation. According to the Erlangen Program, the projective geometry

72

Chapter 2. Affine Geometry and Projective

Geometry

is the study of properties of geometric figures in projective space which are invariant under the projective general linear group. Clearly, r-flats are carried into r-flats by any projective transformation. Thus the property for a geometric figure being a flat and the dimension of a flat are invariant under the projective general linear group. Now let us study the transitivity properties of

PGLn+i(D).

Proposition 2.18: (i) The group PGLn+i(D) PG (n,D).

is doubly transitive on the set of points of

(ii) The group PGLn+i(D) of points of PG\n,D).

is transitive on the set of independent triples

l

(iii) The group PGLn+i (D) is transitive on the set of independent sets of r + 1 points (0 < r < n). (iv) The group PGLn+i(D)

is transitive on the set of r-flats (0 < r < n).

(v) The group PGLn+i (D) is transitive on the set of (n + 2)-subsets of points such that any n + 1 of them are independent. Proof: It is sufficient to prove (iii) and (v), since (i) and (ii) are special cases of (iii), and (iv) is a consequence of (iii) by Proposition 2.15. Let P 0 , P i , ' ' • ,Pr be r + 1 independent points and the coordinates of P t be (a l0 , flti, • • •, din) (0 < i < r). It is easy to see that the following r + 1 points ^

= (l,0,--- > 0), J B 1 = ( 0 , l , - - - , 0 ) , - - - , £ r = ( 0 , 0 , . . . , 0 , l , 0 , - - . , 0 ) ,

where Ei (0 < i < r) is the point whose i-th coordinate is 1 and all the other oordinates are 0's, are independent. If we can find a projective transforma­ tion which carries P 0 , Pi, • • •, P r into J?0, i?i, • • •, Er, respectively, then (iii) will follow. Form the (r + 1) x (n + 1) matrix /

CLQQ aoi

•••

a0n

a\o an

"'

a\n

\ aro ari

\

" - arn )

2.3. Projective Spaces and Projective

Groups

73

It follows from Lemma 2.16 that the above matrix is of rank r + 1. We can supplement n — r more rows to the above matrix such that the obtained matrix

rp _

/

aoo

GOI

* * *

CLOn

I

a

r0

#r0

* * '

arn

\

a>nO

an2

-• •

ann

\

)

is invertible. Then (a z0 , aiu • • •, a^T'1 = et-, i = 0,1, • • •, r, and hence PiT = E{, i — 0,1, • • •, r. This proves (iii). Now let Po> Pi,' •' j Pn+\ be n + 2 points such that any n + 1 of them are independent. It is clear that any n + 1 of the following n + 2 points ^ , = (l,0,---,0),^1 = (0,l,0,.-.,0),--.,^n = (0,0,.-;,0,l), £ = (1,1,...,1) are independent. If we can find a projective transformation which carries P(b A r " ) ^n+i into E0, J5I, • • •, i? n , J5, respectively, then (v) will follow. By (iii), there is a projective transformation T such that PiT = Ei,i = 0,1, • • •, n. Assume that Pn+iT = (x 0 , #i? • • •, xn). Then any n + 1 of the n + 2 points E0, E1, • • •, 2?n, and (x 0 , #i, • • •, xn) are independent. It follows from Lemma 2.16 that x0 ^ 0,^1 ^ 0, • • • , x n 7^ 0. Clearly the projective transformation / xo-1

\

\"

Xn-1 I

leaves each of the n + 1 points 2?0, # i , • • ■, En invariant and carries the point (x 0 , #i, • • •, xn) into £*. This proves (v).

D

74

Chapter 2. Affine Geometry and Projective

Geometry

The coordinates (xo, #i, • • •, xn) of a point P £ PGl(n, D) introduced at the beginning of this section will now be called the homogeneous coordinates of the point P. When x 0 ^ 0, let & = x 0 - 1 x t ( i = 1,2,---jn), then we can choose (l,£i, • • • ,£ n ) to be the homogeneous coordinates of P and call (£i? *' * >£n) the non-homogeneous coordinates of P . Notice that, only those points (x 0 , Xi, • • • , x n ) for which Xo ^ 0 have non-homogeneous coordinates and they will be called finite points. Points with homogeneous coordinates (0, Xi, • • •, x n ) are called points at infinity. The set of points at infinity is a hyperplane in PGl(n,D), whose equation is x0 = 0

and this hyperplane is called the hyperplane at infinity. The finite points of PGl(n,D) are in one-to-one correspondence with the points of AGl(n,D) ( i , 6 , - - - , 6 i ) — > (6,•••>£»)• Consider the projective transformation (x 0 , xi, • • •, x n ) i—► (y0, yw-y

Vn) = (a*, xu • • •, x n )T,

(2.10)

where ^ — (^«i)o
GLn+i(D).

Then n

3/f = SMit" * = 0,l,---,n. i=o If (2.10) leaves the hyperplane at infinity invariant, in other words, if it carries finite points into finite points, then we must have tj0 = 0 for all j ^ 0. If we let rji = y0~lyi homogeneous form

(i = 1,2, • • • ,ra), then (2.10) can be put into the non-

Vi = (*oo + E tjtjo)-^ 3=1

—1

+ E

tjtji)

j=l n

— ^oo (*oi + E ijtji),

i — 1,2, • • •, n.

2.3. Projective Spaces and Projective

Groups

75

Write the above equations in matrix form, we have (7?1,772,- "iVn)

= ^ 0 0 1 ( 6 , 6 , " - , ^ n ) T i + (^00^01, •••,^00^0n),

(2-H)

where 2\ = (tij)i
= 0 for all *v G V},

and called the dual flat of V.

Proposition 2.19 (Principle of Duality): Let W be an r-flat in PGl(n,D), then WL is an ( n - r - l ) - f l a t in PGr(n, D). Let V be an 5-flat in PGr(n, £>), then VL is an (n — 5 — l)-flat in PGr(n1D). Moreover, W—►

WL

is a bijection from the set of flats in PGl{n,D) the following properties: 1° d i m W + d i m W a = n - l , 2°

{WL)L

= W,

to that of PGT(n,D)

with

Chapter 2. AfEne Geometry and Projective

76 3°

Wx DW2if

4° ( ^ n ^ j

and only if WXL C W2L ^ ^ u ^ 1 , {w1uw2)L =

w11nw21

Geometry

□

The proof of this theorem rests on the theory of system of linear homoge­ neous equations in Section 1.4 and is easy. So we leave it to the reader.

2.4

Fundamental Theorem of the Projective Geometry

Clearly, any projective transformation of the n-dimensional left projective space PGl(nyD) over the division ring D carries r-flats into r-flats. In particular, it carries lines into lines. Now we would like to characterize the projective transformations with the sole property that they carry lines into lines. At first, we have the following lemmas. L e m m a 2.20: Let D be a division ring, n be an integer > 1, and A be a bijective map of PGl(n,D) to itself carrying lines into lines. Then ^ _ 1 a l s o carries lines into lines. □ The proof of Lemma 2.20 is the same as that of Lemma 2.8 and hence is omitted. L e m m a 2.21: Let D be any division ring, n > 1, and A be a bijective map of PGl(n,D) to itself carrying lines into lines. Then both A and A'1 carry also r-flats into r-flats for any r = 0 , 1 , • • •, n. Proof: We apply induction on r to prove this lemma. In virtue of Lemma 2.20, it is enough to prove the lemma for A. By hypothesis the lemma is true for r = 0 and 1. Let r > 2, then n > 2, and assume that the lemma is true for r — 1. That is, if A is a bijective map of PGl{n, D) which carries lines into lines, then A carries also (r — l)-flats into (r — l)-flats. Let F be an r-flat in PGl{n, D), and let U be the (r + l)-dimensional subspace of the (n + 1)dimensional row vector space 2)( n+1 ) corresponding to F. Choose a basis ui, i/2, • * •, u>r+i of U. Let UT = < wi, ^2, * • *, ur > and W = < ur+i >. Then UT is an r-dimensional subspace of D^n+1\ W is a 1-dimensional subspace of D(n+1\ U = Ur + W, and Ur H W = < 0 >. Ur corresponds to an (r - 1)-

2.4. Fundamental Theorem of the Projective

Geometry

77

flat F r _i of PGl{n,D) and W is a point of PGl{n,D). Clearly F r _ ! C F l and W G F. Let W = A(W), then W is a point in PG (n,D) and hence is a 1-dimensional subspace of Z)( n+1 ). Let F r '_ x = ^ ( P ^ x ) , by induction hypothesis F^_x is an (r - l)-flat of PGl(n, D). Let E^ be the r-dimensional subspace of Z}(n+1) corresponding to Fj.^. Clearly Ufr C\ W = < 0 >. Let U' = U'r + W , then U' is an (r + l)-dimensional subspace of P>(n+1). Let F' be the r-flat corresponding to U'. We assert that A(F) = F'. Let P be any point of F , then P is a 1-dimensional subspace of U. If P C C/r as a subspace, then P G P r - i as a point and hence A(P) G A(Fr-i) C 4 ( F ) . If P = W, then 4 ( P ) = A(W) C 4 ( F ) . Now consider the case that P £ C/r and P ^ W. Let P = < u >, where v G P)( n+1 ) and t; ^ 0, then we can decompose v into v = u + w, where u G Ur,w G W, u ^ 0, and w ^ 0. Then the point P lies on the line joining the points < u > G P r - i and < K; > = W. By hypothesis 4 ( P ) lies on the line joining the points A(< u >) G 4 ( F r _ i ) = P;_! and 4 ( ^ 0 = W. Let 4 ( P ) = < < / > , then v1 = u' + w\ where u' € U^w' e W. Hence v' G (7' and 4 ( P ) G F ' . Therefore we have proved that 4 ( F ) C F ' . In a similar way we can prove that F' C A(F). Hence .4(F) = F ' . □ Corollary 2.22: Let J9 be any division ring, n > 2, and .A be a bijective map of PGl{n,D) to itself which carries lines into lines. Then for any r = 1,2, • • •, rz, A carries any set of r + 1 independent points also into a set of r + 1 independent points. Proof: Let Pi, P 2 , • • •, P r +i be r + 1 independent points of PGl(n, D). Then they do not lie on any (r-l)-flat. By Lemma 2.21 4 ( P i ) , 4 ( P 2 ) , • • •, A(Pr+i) also do not lie on any (r - l)-flat. Therefore 4 ( P i ) , 4 ( P 2 ) , • • •, A(Pr+i) are r + 1 independent points. Q Now we can state and prove the fundamental theorem of the projective geometry over any division ring. Theorem 2.23: Let D be any division ring and n be an integer > 2. Then any bijective map A of the ra-dimensional left projective space PGl(n, D) to itself which carries lines into lines is of the form 4 ( x 0 , x1, • • •, xn) = (x0i xu • • •, xn)aT

(2.12)

Chapter 2. Affine Geometry and Projective

78

Geometry

for all (zo,£i, * * • ->Xn) £ PGl(n,D), where a is an automorphism of D and T e GLn+i(D). Conversely, any map of the form (2.12) from PGl(n,D) to itself is bijective and carries lines into lines. Proof: Let ^

= (l,0,-.-,0),E1 = (0,l,0,...,0),...,En = (0,.-.,0,l), £ = (1,V--,1),

then E0, E\, • • •, En, E are n + 2 points of PGl(n, D) such that any n + 1 of them are independent. By Corollary 2.22, A(E0),A(E1), •• -,A(En),A(E) l are also n + 2 points of PG (n, D) such that any n + 1 of them are indepen­ dent. By Theorem 2.18(v) there is a projective transformation which carries A(E0),A(E1)r--,A(En),A(E) into E0,EU •••, En,E, respectively. Thus after subjecting A to this projective transformation, we can assume that A{Ei) = Ei i = 0 , l , - - - , n , and A{E) = E. The hyperplane at infinity x0 = 0 is the (n — l)-flat containing the points E\,E
1 a 0 Ti

for all (£i,#2, *' * ->xn) £ E, where a is an automorphism of D, a is an ndimensional row vector, and T\ £ GLn(D). From A(EQ) — EQ we deduce

2.4. Fundamental Theorem of the Projective Geometry

79

a = 0, and from A(E) = E we deduce I \ = I. Thus after subjecting A to the bijective map of the form (2.12) (x 0 , xi, x 2 , • • •, x n ) '—► (s 0 , xu x 2 , • • •, z n ) a

,

we can assume that .4(1, xi, x 2 , • • •, x n ) = (1, xi, x 2 , • • •, xn)

for xi, x 2 , • • •, xn e D.

(2.13)

Similarly, for i = 1,2, • • •, n, we have ,4(30, *' *, zt-_i, 1, Zt+i, ■•■,!„) = (x 0 , • • •, xt-_i, 1, xt-, • • •, xn)**,

(2.14)

where 2, comparing (2.13) and (2.14) we obtain CTj = a2 = • • • = an = 1.

Therefore 4 ( x 0 , xi, • • •, xn) = (x 0 , xi, • • •, xn)

for all (x 0 , xi, • • •, x n ) G PGl(n,

D).

□ Theorem 2.23 can be generalized as follows. T h e o r e m 2.24: Let D and D ' be division rings, and n and nf be integers > 2. Let A be a bijective map from the n-dimensional left or right pro­ jective space Pn = PGl(n,D) or PGr(n,D) over D to Pnt = PGl(n,Df) or PGr(n,D'). Assume that A carries lines into lines. Then n = n' and D is isomorphic to D' or anti-isomorphic to D'. In the first case, Pn and Pni are either both left projective spaces or both right projective spaces. If Pn = PGl(n, D) and Pn> = PGl(n, D'), then 4 is of the form 4 ( x 0 , xi, • • •, x n ) = (x 0 , xi, • • •, xn)aT for all (x 0 ,*i, • • • ,xn)ePGl{n,D); then 4 is of the form

if P n = PGr{n,D)

and P n , = P G r ( n , D ' ) ,

4*(X 0 , Xi, • • • , Xn) = T[*(X0, «!,•••» Xn)Y

80

Chapter 2. AfBne Geometry and Projective

Geometry

for all t(xo,Xi,- — ,xn) G PGr(n,D)', in the above two formulas a is an isomorphism from D to D' and T G GLn+i(D'). In the second case, one of Pn and Pni is a left projective space and the other is a right projective space. If Pn = PGl(n, D) and Pn, = PGr(n, D'), then A is of the form A(x0, xu • • •, x„) = T[*(xo, z i , • • •, xn)]T for all ( s 0 , x i , • • - , * » ) € P G ' ( n , D ) ; if P n = PGr{n,D) then .4 is of the form .4 *(so,a:i, •••,£„) = ( x 0 , ^ i , - - -

and P n , =

PGl{n,D')

,xn)TT

for all (xo, a?i, • • •, £ n ) G PGr(n, D)\ in the above two formulas r is an antiisomorphism from D to D' and T G GLn+i(D'). □

2.5

One-dimensional Projective Geometry

When n = l, PGr'(l, D) is simply a line and we call it the left projective line, or the projective line. Clearly, any bijective map from PGl(l,D) to itself carries the projective line to the projective line. So, in order to characterize the projective transformations of PGl(l,D) more invariants are needed. We know from Proposition 2.18(i) that PGL2{D) is doubly transitive on the set of points of P G ' ( 1 , D). Moreover, Proposition 2.18(v) can be interpreted as that PGL2(D) is triply transitive on the set of points of PGl(l,D). But for four points a new invariant called the harmonic set should be introduced. Let (#o, xi) be a point of PGl(l, D). If x0 ^ 0, then it is called a finite point whose non-homogeneous coordinate is XQ lx^. If x0 = 0, then it is called the point at infinity and let oo be its non-homogeneous coordinate. We agree that for any a, 6, c, d G D, (a + oo • c) _1 (6-|- oo • d) = c " 1 d , a " 1 ( 6 + oo • d) = oo, (a + oo • c)~1b = 0, (a + b • oo)(c + d • oo)" 1 = W 1 , (a + 6 • oo)c _ 1 = oo, a(c + d • oo)" 1 = 0, a • 0" 1 = 0" 1 • a = oo,

2.5. One-dimensional Protective Geometry oo-a-oo

1

81

= oo * • a • oo = a.

Consider the protective transformation 'a (
d


where A G D * and

fa b>

b\ j ,

d)>

(2.15)

(:iH«">\c

dj

| €

GL2(D).

In non-homogeneous coordinates, (2.15) can be expressed as (a ++ sxc)~ xd). y == (a c ) " l1(b ^ ++ xd), y--

(2.16)

w

X = where yt/ == y^yi = aXQ^ Z i - Let 2/0 V and x

faa b^ bY1I

\c x c dd}j1

- 1((a'a' V b'\

~ \[c' c' d'd'j-

" I

Then \ _ _fa> ( a' bb'' \\ fa ( a 6b\ \ _ fl / 1 00\\ faa bb\\ ( fa'a' d VV\_ \c c d)\d d) U ')~ d' ) ~"U \c' d' ){c d ) ~ \0 *) [c dj--{0 l1 j) -'

(2.17)

Using the above identity, it can be readily verified that (a + xc)(-b'-&' + ax) ax) -== (b(b++xd)(d xd)(d'f --- c'x). + xc)(Then we also have 1 1 (-&' + a'x){d' a'x)(d' -- c'x). . yy = (-V c'x)-

(2.18)

Definition 2.1: Let X\,x2,x3 and x4 be four points on the projective line PGl(\,D) in non-homogeneous coordinates. They are said to form a har­ monic set, if (x11 --- x£44) == —1. -1. - x3)(x1 ---xx3)3)~11(x ) 11(x (x22 -(x2 --- x4)~

(2.19)

□ l

We want to characterize those bijective maps of PG (l, D) which carry har­ monic sets into harmonic sets. At first we give the following definition.

82

Chapter 2. AfRne Geometry and Projective

Geometry

Definition 2.2: Let D and D' be division rings, a bijective map cr : a —> aa(a € D) from D to D' is called semi-isomorphism from D to £)' if for any a,b E D the following conditions hold: {a-rbY

=aa + b%

(2.20)

(a&a)* = a'b'a*,

(2.21)

and r = 1. When D — D', it is called a semi-automorphism

(2.22) of D.

□

Set b = 1 in (2.21) and (2.22), we deduce (a2)* = (a*) 2 .

(2.23)

We have the following beautiful theorem of Hua. Theorem 2.25 (Hua): Every semi-isomorphism from a division ring to another division ring is either an isomorphism or an anti-isomorphism. In particular, every semi-automorphism of a division ring is either an automor­ phism or an anti-automorphism. Proof: Let D and D' be division rings and a : a —» a°(a £ D) be a semiisomorphism from D to D'. For any a,b,c £ D, from (2.20) and (2.21) we deduce (abc + cba)a = ((a + c)b(a + c) - aba - cbcf = (a* + ca)ba(a° + ca) - aabaaa - cabac° = aabaca +cab°a°.

(2.24)

Then from (2.20)-(2.24) we deduce {(abY - a*b") ((aby - b°a°) = ((abY)2 + a°(V)2a* - (a°V(aby + (ab)aWa') = ((ab)2)" + (ab2af - (ab(ab) + (ab)ba)' = 0. Since D' is a division ring, we have (ab)a = aaba or baaa for all a, b £ D. If D' is a field, then, clearly, Theorem 2.25 is true. Now let D' be not a

2.5. One-dimensional Projective Geometry

83

field, i.e., the multiplication of D' is not commutative. We distinguish the following two cases: (a)

There is a pair of elements a, b G D' such that (abY = aaba ±

¥aa.

We prove by the method of "infection" that for any c,d G D (cdy

=cad%

i.e., cr is an isomorphism from D to D'. At first we assert that (cb)a = caba and (acy = a° ca. Suppose that there is a c G D such that (cb)a = baca ^
There is a pair of elements a, b G -D such that (ai)* = fc'a' ^ aaba.

Then we can prove in a similar way that for all c,d € D (cd)a

=dac°,

that is, <7 is an anti-isomorphism from D to D'. Let D be a division ring, cr be a semi-automorphism of D, and

(•j)£MlW.

□

84

Chapter 2. Affine Geometry and Protective

Geometry

Then the transformation of PGl(l, D) of the form X i H\—> j /y

a

1 1

x*d) x c)" ■ (b (b++x°d) fl -|=== ((a + x"c)-

is called a generalized projective transformation.

We have

Theorem 2.26: Generalized projective transformations of the projective line PGl(l,D) carry harmonic sets into harmonic sets. Proof: By definition a generalized projective transformation is the product (or composite) of a semi-automorphism and a projective transformation. We discuss these two cases separately. (a) Let x X

— i\—> ►2/y=== (a (a++xc)~ £c)■-11(& xd) (b+ +xd)

be a projective transformation of the projective line. By (2.16), (2.17), and (2.18), we have Vi ~~ 2/j Vj = (a + Xic)'1^

+ Xid) - (-&' + a'xj){d' a'xj)(d' -

C'XJ)-1

= (fl + Xic)- ^ + Xid)(d' - C'XJ) - (a + Xic)(-b' Xic){-b' + a'xj)){d' -- J^Y C'XJ)'11 x1 1 (x{ — Xj)(d' — c'xj)' . = (a + Xic)~ (xi 1

Thus (2/2 - y*) 2/4)"x1(j/2 (2/2 - y3){yi 2/3X2/1 - yz) 2/3)~11(yi (2/i -- y*) 2/4) (y2 1 1 = (df (df -—c'x c'x4)(x x3)(xi- — x4)(d' c,x14.)~1. = (x2(x 2- — x3)(xi x3)~x31)~ (xi1(xi-- x— -- —c'xi)4)(x 4 )~ 22 -—x4x)~ 4)(d'

Hence 2/1,2/2,2/3,2/4 form a harmonic set if and only if X\) #2? X3l X4 form a harmonic set. (b)

Consider the transformation a x \-y H-» y = = xx°, ,

(2.25)

where a is a semi-automorphism of D. By Theorem 2.25 a is either an automorphism or an anti-automorphism. If a is an automorphism of D, then clearly (2.25) carries harmonic sets into harmonic sets. Let a be an anti-automorphism of D. We have the identity l _ 1l a-\a a~\a ± 6)6" 6)6- x1 = b~ b'\a (a ± 6)a b)a~ for all a,b a,be€ D*. D\

(2.26)

2.5. One-dimensional Projective Geometry

85

Suppose that xi,x2jx3,x4 form a harmonic set, then we have (2.19). Mul­ tiplying both sides of (2.19) by (x2 - x1)~1(x2 - x4) from the left and by (xi — x4)~x from the right, we obtain (x2 - x1)~1(x2 - z 3 )(xi - X3)"1 = -(x2 - x1)'1(x2

- x4)(xi - X4)"1. (2.27)

Applying (2.26) to both sides of (2.27), we get (xi - x3)~1(x2

- x3)(x2 - X4)"1 = -(xx

- x4)-1(x2

- x4)(x2 - x i ) " 1 .

Multiplying (2.27) by (xx - x4) from the left and (x2 - xi)(x2 - X4)"1 from the right, we have (xi - x4)(xi - x3)~1(x2 - x3)(x2 - X4)"1 = - 1 .

(2.28)

Applying a to (2.28) we finally obtain \x2

That

ID,

U/-I

~

,«t/n,«{/o,

X

4J

\X2

~

X

3J\X1

~

X

3J

\X1

~

X

4)

=

~~■*■•

and x\ form also a harmonic set.

□

Conversely, we have the fundamental theorem of the one-dimensional pro­ jective geometry over any division ring. Theorem 2.27 (Hua): Any bijective map of the left projective line over a division ring of characteristic not two, which carries harmonic sets into harmonic sets, is a generalized projective transformation. Proof: Let D be a division ring of characteristic not two, A be a bijective map from PGl(\, D) to itself, and assume that A carries harmonic sets into harmonic sets. ,4(oo), -4(0), and -4(1) are three distinct points. By Proposi­ tion 2.18(v) there is a projective transformation which carries them into 00, 0, and 1, respectively. After subjecting A to this projective transformation we can assume that ,4(oo) = 00,

-4(0) = 0, and .4(1) = 1.

For simplicity we write A(x) = xa', then 00* = 00,

oa = 0,

r = 1,

86

Chapter 2. Affine Geometry and Projective

Geometry

and the bijective map a : x h-> xa carries harmonic sets into harmonic sets. We want to prove that a is a semi-automorphism of J9, then Theorem 2.27 will be proved. Let x and y be any two elements of Z?, and let = 2x, 2x, xx2 == 2t/, 2», X\ = 2 Xi

and

1, x33 === ^{xi 2{Xl + + x22)-).

(2.29) (2.29)

Then -xs)-11 0*2-33)0*1 0*2 -- x3){xi -Xs)'

= = -- 11 ..

Thus x i , x 2 , ^ 3 , and 00 form a harmonic set. Since a carries harmonic sets into harmonic sets, x^,x2,x3, and 00 form a harmonic set. Therefore we have — x3)(x1 —- *£3) [x2 — #3)(2?i K! ) - 1 == - 11-. Thus

1

*l 3 _== \(*l xl). -2W+ *Z)-

.

x

2 ((2.30) -30)

From (2.29) and (2.30) we deduce ^

1

(gfa +**))*

=

X ^ i "*" a52)>

i.e., (x + yr = i ( ( 2a; 2 xrr - ( 22yy)r' ) .. (* + »)* = 2(( Let y = 0 in the above equation, then we have a

X x f f :=

-(2xY.

; = 25'V 2l»"'

Therefore a a (* (x + y)a = y)x" =x+a+y y°-.

Let X\ = x,x2 = 1 — x, and £3 = 2x\x2^ then Xi,£2?#3? and 0 form a harmonic set. Therefore x^x2,xl, and 0 form a harmonic set, i.e., (x [x2)

[x22 —xx3)(x [x 1l 3)[x

—* * x3 /3)

X

l

x 1~~ = —1. *■'

2.6.

Comments

87

It follows that £3 = 2x\xa2. Hence (2x(l-x)Y

2xa(l-x)%

=

from which follows

(*2r = K) 2 . Thus

(xy + yxY = ((x + y)2 - x2 - y2f = (** + y * ) 2 - ( * * ) 2 - ( y * ) 2 = x*ya + yax*.

From 2xyx = x(yx + xy) + (yx + xy)x - (x2y + yx2), we deduce {xyxY =

x°y°x°.

Therefore a is a semi-automorphism of D.

2.6

D

Comments

The presentation of the affine space and projective space over any division ring in Section 2.1 and 2.3, respectively, are standard, and we follow mainly Wan 1993. Section 2.2 is adopted from Hua 1951, but with modifications and correc­ tions. The fundamental theorem of the affine geometry over the field of real numbers R can be found in Veblen and Whitehead 1933 and that over any division ring D appeared in Hua 1951. But for the case of dimension n > 3 and D = ¥2 the hypothesis of the theorem should be modified, which was pointed out in Wan 1961. The fundamental theorem of the projective geometry over any division ring is a rather old result and can be found in many books, e.g., Baer 1952 and Artin 1957. In Section 2.4 we derive it from the fundamental theorem of the affine geometry over any division ring, which is proved in Section 2.2. If one likes, one can also prove the fundamental theorem of the projective geometry over any division ring directly and then deduce the fundamental theorem of the affine geometry over any division ring from it.

88

Chapter 2. Afiine Geometry and Projective

Geometry

All the beautiful results and elegant proofs in Section 2.5 are due to L.K.Hua, for Theorem 2.25, cf. Hua 1949 and for Theorem 2.26 and 2.27, cf. Hua 1951.

Chapter 3 Geometry of Rectangular Matrices 3.1

The Space of Rectangular Matrices

Throughout this chapter let D be a division ring and m, n be integers > 2. Denote the set o f m x n matrices over D by M.mXn(D). When m = n, we write simply Mn(D) for Mnxn(D). MmXn(D) is the space we are going to study in this chapter and we call it the space ofmxn matrices over D, or simply the space of rectangular matrices if the size of the matrices is clear from the context. We also call the matrices in MmXn(D) points of the space. With the space M.mxn(D) we associate naturally a group of motions which consists of transformations of the form X H—> PXQ + R

for all X G Mmxn{D),

(3.1)

where P G GLm(D), Q G GLn{D), and R G Mmxn(D). Clearly, (3.1) is bijective. Denote this group by GmXn(D). We begin with the study of the transitivity properties of the group GmXn{D) acting on the space Mmxn{D). Proposition 3.1: The group GmXn(D)

acts transitively on

MmXn{D)-

Proof: Let X\ and X2 be any two m x n matrices over D. transformation X i—► X + (X2 — X\) carries X\ to X2.

Then the a

Definition 3.1: Let Xi, X2 G Mmxn(D). They are said to be of arithmetic distance r, denoted by ad(Xi, X2) = r, if rank(Xi — X2) = r. When r = 1, 89

90

Chapter 3. Geometry of Rectangular

then they are said to be adjacent (or coherent).

Matrices □

The arithmetic distance fulfills the three requirements for the distance func­ tion in a metric space. Proposition 3.2: Let Xu X2, X3 G Mmxn(D).

Then

1° ad(Xi, X2) > 0; ad(Xu X2) = 0 if and only if X1 = X2. 2° a d ( X i , X 2 ) = a d ( X 2 , X i ) . 3° ad(X l 5 X 2 ) + ad(X 2 , X 3 ) > ad(Xi, X 3 ). Proof: 1° and 2° are clear, and 3° follows from Proposition 1.21.

□

Proposition 3.3: The elements of the group GmXn(D) leave the arithmetic distance between any pair of points of M.mxn{D) invariant. Moreover, for any r with 1 < r < min{m, n } , the set of pairs of m x n matrices over D of arithmetic distance r forms an orbit under Gmxn(D). Proof: The first statement is clear. Let us prove the second statement. Let Xi, X2 be a pair o f m x n matrices over D which are of arithmetic distance r, i.e., rank(Xi — X2) = r. Let R be an m x n matrix of rank r. It is enough to show that there is an element of Gmxn(D) which carries X\ and X2 to 0 and R, respectively. Clearly, the transformation X i—> X — X\ carries X\ to 0 and X2 to X2 — X\. We have rank(X 2 — Xi) = r. Since R is also of rank r, by Corollary 1.19 there is an element P G GLm(D) and an element Q G GLn(D) such that P(X2 — X\)Q = R. Clearly, the transformation X i—► PXQ leaves 0 invariant and carries X2 — X\ to R. D Therefore the arithmetic distance between any pair of points of MmXn(D) is a geometric invariant under the group Gmxn(D), so is, in particular, the adjacency of a pair of points of M.mxn{D). We would like to characterize the elements of the group GmXn(D) by as few geometric invariants as pos­ sible. We will see that the invariance of the adjacency of pairs o f m x n matrices alone is sufficient to characterize the transformations of the form (3.1) to within automorphisms of D if m ^ n. More precisely, we have the following fundamental theorem of the geometry of rectangular matrices over any division ring.

3.1.

The Space of Rectangular

Matrices

91

Theorem 3.4: Let D be any division ring, m and n be integers > 2, and A be a bijective map from Mmxn(D) to itself. Assume that both A and A'1 preserve the adjacency of pairs of m x n matrices, i.e., for any X i , X2 G Mmxn(D), X\ and X2 are adjacent if and only if A(Xi) and A(X2) are adjacent. Then when m ^ n, A is of the form ,4(X) = PJSTQ + R

for all X G Mmxn{D),

(3.2)

where P G GLm{D), Q G GLn(D), ReM mx.n{D), cr is an automorphism of D, and X a denotes the matrix obtained from X by applying a to all the entries of X. When m = n, in addition to (3.2), we have also A(X)

= P\XT)Q

+R

for all X G MmXn(D),

(3.3)

where r is an anti-automorphism of D. Conversely, any map of Mmxn(D) to itself of the form (3.2) (and of the form (3.3), when m = n) is bijective, and both the map and its inverse map preserve the adjacency of pairs of m x n matrices. After some preparations in Sections 3.1 - 3.3, the proof of Theorem 3.4 will be given in Section 3.4. Definition 3.2: Let X, X' G Mmxn(D). When X ^ X', they are said to be of distance r, denoted by d(A", X') = r, if r is the least positive integer for which there is a sequence of r + 1 points Xo, «X"i, • • •, Xr with XQ = X and Xr = X' such that X{ and Xi+i are adjacent, z = 0, 1, • • • , r — 1. When X = X', we define d(X, X) = 0. □ Proposition 3.5: For any two points X, X' G A^ m X n(^)j ad(X, X') = d(X,

X').

Proof: When X = X', clearly we have ad(X, X ) = d(X, X) = 0. Now let X ^ X'. Assume that ad(X, X') = r, i.e., rank(X - X') = r. By Proposition 1.17 there is an element P G GLm(D) and an element Q G GLn(D) such that P(X-x')3=(/(r)

0(m_r,n_r)).

Chapter 3. Geometry of Rectangular

92

Matrices

Let Hi = P-1 ( 7

%

0 ( r o - t -. n - 0

) Q~\

i = 1, 2, • • ■, r.

Then Xo

=

-X"? -Xi = X — i?i, X2 = A. — it25 * * * > XT = X — Rr = X

is a sequence of r + 1 points of MmXn(D) such that X{ and X,+i are adjacent, * = 0, 1, 2, • • •, r - 1. Therefore d(X, X') < r = ad(X, X ' ) . Conversely, suppose that d(X, X') = r'. Then there is a sequence of r' + 1 points X 0 , Xi, X2, • • •, Xrt with Xo = X and J£r/ = X ' such that X{ and X t + i are adjacent, i = 0, 1, 2, • • •, rf — 1. By Proposition 1.21 d(X, X') = r' = r £ * rank(Xt- -

Xi+1)

t'=0

> rank(X 0 - Xr.) = rank(X - X') = ad(X, X ' ) . Therefore ad(X, X') = d(X, X'). D

Corollary 3.6: Let A be a bijective map from MmXn(D) to itself and as­ sume that both A and A'1 preserve the adjacency of pairs oimxn matrices, then A also preserves the arithmetic distance between any pair oimxn ma­ trices, i.e., for any X, X' G Mmxn(D), ad(X, X') = *d(A(X), A{X')). Proof: Suppose that A preserves the adjacency of pairs oimxn i.e., for any Xi, X2 G Mmxn(D), adjacency of A(Xi)

and A(X2).

matrices,

the adjacency of X\ and X2 implies the Then for any X, X' G

d(X, X') > d{A{X),

A{X')).

By Proposition 3.5 we have ad(X, X') > ad(A(X),

A{X')).

Similarly, if A~x preserves also the adjacency, then a d ( ^ ( X ) , A{X'))

> ad(X, X').

MmXn(D),

3.2. Maximal Sets of Rank 1

93

Therefore, if both A and A 1 preserve the adjacency, then for any X, X' € Mmxn(D), ad(X, X') = ad(A(X), A(X')).

□

3.2

Maximal Sets of Rank 1

Definition 3.3: Let M. be a non-empty set of points in the space M.mXn(D). M. is said to be a maximal set of rank 1, if any two points of M. are adjacent and there is no other point outside M., which is adjacent to each point of M. D Proposition 3.7: A maximal set of rank 1 in AimXn{D) is carried into a maximal set of rank 1 under any transformation of the form (3.1). Proof: It follows from Proposition 3.3.

□

Proposition 3.8: Both f/Zll

Mi = I

#12

0

0

0 E n , ^12, • • • , xln

Vo

e

D

(3.4)

0 /

o

and ( J/u

M[ =

2/21

0 0

0\ 0 S/ii, 2/2i, • • • , 2/mi € D

\J/ml

0

(3.5)

0)

are maximal sets of rank 1. Moreover, every maximal set of rank 1 can be carried under a transformation of the form (3.1) to either Ai\ or M.[. Proof: Clearly, any two points of M\ (or M[) are adjacent. That there is no other point outside .Mi (or Ai[) which is adjacent to every point of M.\ (or M!x, respectively) will follow from the argument of next paragraph. Let M. be a maximal set of rank 1 contained in A4mXn(D). Clearly M. cannot consist of a single point. Let X and X' be a pair of points in A4, then they are adjacent. By Proposition 3.3 there is a transformation of

Chapter 3. Geometry of Rectangular

94

Matrices

the form (3.1) which carries X and X' into 0 and E n , respectively. The transformation will carry M into a maximal set of rank 1 containing 0 and E\\. Thus we may assume that M contains 0 and En. Let X\ be another point of M. Since X\ and 0 are adjacent, X\ is of rank 1. Therefore we may write ( Gl&l

a2bi *i

CL2b2

0-2K

=

\ambi

amb2

where ai, a2, • • •, am 6 D and are not all zero, and &i, &2> • • •, bn € D and are not all zero. Since X\ and En are also adjacent, X\ — En is of rank 1. Therefore aibj = 0, i = 2, 3, • • •, m; j — 2, 3, • • •, n. If 62, 63 thus

, bn are not all zero, then a2 = a 3 = • • • = a m = 0, a\ ^ 0, and (d\bi

0 *i

aib2 0

0

=

V 0

0

(3.6)

0 y

Otherwise, we have

*i

where b\ ^ 0 and a2> a 3 , and

/ ai&i a2bi

0

\a>mb\ 0

0/

=

(3.7)

a m are not all zero. The intersection of M\

JM'X

{xEu

\xeD}

(3.8)

is evidently not a maximal set. If M contains a point outside (3.8), say a point of the form (3.6) with ai ^ 0 and b2, 63, • • •, bn not all zero, then none of the points of M can be of the form (3.7) with &i ^ 0 and a2> «3, • • •, am not all zero. Similarly, if M contains a point of the form (3.7) with h ^ 0 and a2, «3, • • •, «m not all zero, then none of the points of M can be of the form (3.6) with «i ^ 0 and 62, b3, • • •, bn not all zero. Clearly every pair of points in Mi (or M[) are adjacent. Therefore M is either Mi or M J . D

3.2. Maximal Sets of Rank 1

95

Corollary 3.9: A maximal set of rank 1 is either of the form 0

0

0

Q + R xn, a;i2, • • •, Zi„ G D

(3.9)

0 /

\ 0 or of the form / 2/ii 2/21

0 0

where P G GLm(D),

0\ 0

Q + R 2/n, 2/21, ••*, 2/mi € D

Q G GLn(D),

(3.10)

and # G A^ m X n(^).

D

The proof of Proposition 3.8 gives the following corollaries. Corollary 3.10: Given any pair of adjacent points in Mmxn(D), there are two and only two maximal sets of rank 1 containing both of them. □ Corollary 3.11: The intersection of two distinct maximal sets of rank 1 which contains more than one point in common can be carried into (3.8) under a transformation of the form (3.1). □ Definition 3.4: If the intersection of two distinct maximal sets of rank 1 contains more than one point in common, then it is called a line. □ Corollary 3.12: There is one and only one line passing through any pair of adjacent points. □ Corollary 3.13: The parametric equation of a line in AdmXn

is

{lpxq + R\xeD}, where p is a nonzero m-dimensional row vector over D, q is a nonzero ndimensional row vector over D, and R G MmXn(D). The parametric equa­ tion of a line in the maximal set of rank 1 (3.4) is (

/an 0

ai2 0

V o

o

Q>ln\

0

x 0 /

/ &11

+

&12

0

0

0

V o

o

0 /

xeD

Chapter 3. Geometry of Rectangular

96

Matrices

where (an, #12, • • •, Q>\n) 7^ 0, and the parametric equation of a line in the maximal set of rank 1 (3.5) is f / an

0

«21

0

K\aml

^

0

•••

&21

0

Uml

0

x+

0/

x £D 0/

where ' ( a n , a 2 i, • • •, a m l ) ^ 0.

D

Furthermore, we have Proposition 3.14: Two maximal sets of rank 1 which have only one point in common can be carried simultaneously under the group Gmxn{D) to either (3.4) and

i I 0

0 \

0

Xix

Xi2

Z2n

0

0

0

I\ 0

0

o /

#21, #22, '" , X2n € D

(3.11)

2/12, 2/22, • • • , 2/m2 €

(3.12)

or to (3.5) and /0

J/12

0

0

y22

0

V 0 ym2

0\ 0 D

0

Proof: Let A4 and M! be two maximal sets of rank 1 which have only one point in common. By Propositions 3.1 and 3.8 we can assume that the common point of M and M! has been carried to 0 and M has been carried into (3.4) or (3.5) by an element of G m x n ( J D). Consider the case when M has been carried into (3.4). (The other case can be treated in a similar way.) Let M! be carried into M" and let

X2 =

«21

V «ml

«12

«ln \

«22

Q>2n


3.3. Maximal Sets of Rank 2

97

be an element of M" distinct from 0. Since X2 is not an element of (3.4), the submatrix formed by the last m — 1 rows of X2 is of rank 1. Thus there is an element P G GLm^1(D) and an element Q G GLn(D) such that C12

•••

1 0

0 0

••• •••

V0

0

/ cu

P)X*Q=

0

cln\

0 0

(3.13)

o )

The above matrix is of rank 1, so c i 2 = • • = Cin = 0. Notice that the transformation

*^0

p)XQ

leaves 0 invariant and also the maximal set of rank 1 (3.4) invariant. Then the transformation /l

-en 1

\ 1

X \

X 1/

leaves (3.4) invariant and carries (3.13) to £"21. By Corollary 3.10 there are exactly two maximal sets of rank 1 containing 0 and £21 • Clearly, (3.11) and (3.5) are two maximal sets of rank 1 containing 0 and £21 • Since (3.4) and (3.5) have more than one point in common, (3.5) is impossible. Therefore we must have (3.11). Q

3.3

Maximal Sets of Rank 2

Definition 3.5: Let C be a set of m x n matrices over D. C is called a maximal set of rank 2, if the following conditions are fulfilled: 1° C contains two maximal sets of rank 1 which have only one point in common. Denote these two maximal sets of rank 1 by M and M\ and their point of intersection by A. 2° C contains all the points which are of arithmetic distance 2 from A, of arithmetic distance < 2 from every point of M. and M!\ not of arithmetic

Chapter 3. Geometry of Rectangular

98

Matrices

distance 2 from every point of JM, and not of arithmetic distance 2 from every point of M!. Denote the set of these points by M. 3° C contains all the points which are of arithmetic distance 1 from A, of arithmetic distance < 2 from every point of Af, and not of arithmetic distance 2 from every point of N. 4° C contains no other points.

D

Proposition 3.15: Any maximal set of rank 2 in Mmxn(D) is carried into a maximal set of rank 2 under any transformation of the form (3.1). □ Proposition 3.16: Both /#11

X12

X2\

X22

#2n

0

0

0

IV 0

0

o /

#11, #12, ' * , # l n , #21, #22, ' * * , #2n € D

(3.14)

and f / 2/ii

2/12

0

2/21

2/22

0

o y i l , 2/21, • • • , J/ml, 2/12, 2/22, • • • , V-m.1 S -D

I \ 2/ml

2/m2

0

(3.15)

0/

are maximal sets of rank 2. Moreover, any maximal set of rank 2 can be carried under the group Gmxn(D) into either (3.14) or (3.15). Proof: Let us prove the second statement. The first statement will be proved simultaneously. Let £ be a maximal set of rank 2. By Definition 3.5 C contains two maximal sets of rank 1, Ai and M!\ which have only one point in common. By Proposition 3.14 we can assume that Ai is (3.4) and M' is (3.11), or M is (3.5) and M' is (3.12). We consider only the first case, since the second case can be treated in a similar way. If m = 2, then obviously our proposition holds. So in the following we assume that m > 3.

3.3. Maximal Sets of Rank 2

99

Since M 0 M' = { 0 }, we have A = 0. Let / #11

X12

#ln

^21

^22

Z2n

\ Xmi

Xm2

X =

\

be any point in M. Then ad(X, 0) = 2, i.e, rank X = 2. Denote X," =

^X t 'i, Xi2)

? '^inj'i

1

=

I?

^)

ra.

Since X is of arithmetic distance < 2 from every point of M. and not of arithmetic distance 2 from every point of M, we have (X2\ rank

= 1.

Vx m / Similarly, rank

= 1.

If there is some i (3 < i < m) such that xt- ^ 0, then there exists Ik £ D (k = 1, 2, • • •, m) such that ra.

#A; — 'fc^n ' & — -•-> A

It follows that r a n k X = 1, which is a contradiction. Therefore X{ = 0 for i = 3, 4, • • •, ra, and X is a matrix of rank 2 of the form

^21

#22

%2n

0

0

0

V 0

0

o )

All matrices of rank 2 and of the above form constitute M.

Chapter 3. Geometry of Rectangular

100

Matrices

Let 2/12

2/m\

2/21

2/22

2/2n

\ 2/ml

2/m2

(2/n

Y =

* '*

2/mn /

be a point which is of arithmetic distance 1 from 0, of arithmetic distance < 2 from every point of Af, and not of arithmetic distance 2 from every point of Af. Then r a n k F = 1. Denote (Vi = 2/;i, 2/i2, " • ' , 2/m),

i = 1, 2, • • •, m.

If y3 zfi 0, then there exists Ik G D (k = 1, 2, 4, 5, • • •, m) such that Vk = 42/3, fc = 1, 2, 4, 5, •••, m. The transformation /l

-h ~/ 2 1 -U

1 X

\ X

l

carries A^i, M.2, and AT into themselves, and carries Y to 0

y* =

2/3

0

Vo/ Then Y* is of arithmetic distance > 2 from every point of Af, which is a contradiction. Therefore y3 = 0. Similarly y4 = y5 = • • • = ym = 0. Hence F is a matrix of rank 1 of the form 2/12

fyn

2/21

2/22

0

0

\ 0

0

•••

2/in\

2/2n

0

3.3.

Maximal

Sets of Rank

2

101

Therefore C is (3.14). C o r o l l a r y 3 . 1 7 : A m a x i m a l set of rank 2 is either of t h e form #12

•'•

#21

#22

''*

X2n

0

0

•••

0

V0

0

•••

0 /

IXX\

Xln\

Q+R

# i i , #12, • • *, xin, ^ 2 1 , £22, • • • , x2n

G i?

or of t h e form f

[ Vu T/21

I

\yml

2/12

0

•••

0\

2/22

0

•••

0

Vm2

0

•••

where P G GLn(D),

g + i? 2/ll, 2/21, • ' * , 2/ml, 2/12, 2/22, * * * , 2/m2 G £>

0)

Q G GLn{D),

a n d i? G M m x n ( 5 ) .

°

Now let us s t u d y t h e intersection of a m a x i m a l set of rank 1 and a m a x i m a l set of rank 2, when it is nonempty. We can assume t h a t their intersection contains 0. By Corollary 3.9, a m a x i m a l set of rank 1 containing 0 is either of t h e form /Xn

#12

A°°

0

l

0 /

\ 0

0

Q

£ n , ^12, • • *, xln

G D

or of t h e form

2/21 \ymi where P € GLm(D),

0\ 0

0

Q

yn, 2/21, • • • , ym\ € D

0)

0 Q G GLn(D).

By Corollary 3.17 a m a x i m a l set of rank

2 containing 0 is either of t h e form Xl„\ X21

X22

X2>

0

0

0

I Vo o

o )

Q

X\l,

X12, • • • ,. X i „ , 1 2 1 , 222, • • • , X2n € £>

Chapter 3. Geometry of Rectangular

102

Matrices

or of the form f

[

/ 2/ii

2/12

0

•••

0\

2/21

2/22

0

•••

0

KVml

2/m2

0

•■■

Oj

Q

2/11? 2/21, • • • , 2/ml, 2/12, 2/22, * * • , 2/m2 €

£

where P G GLm(D) and Q G GLn(D). Therefore we can assume that the maximal set of rank 1 containing 0 and the maximal set of rank 2 containing 0 are carried under a transformation of the form (3.1) into one of the following four cases: (a)

r / yn p

f/xn 0

X12 ••• 0 •••

xln\ 0

IV o

o

0 /

yi2

•••

2/21

2/22

• ••

2/2n

0

o

•••

0

\ 0

0

Xu,

X12, ■•• , Xin €

D

yin\

Q

2/11, 2/12, • • • , 2/ln, 2/21, 2/22, • • • , 2/2n € D

o /

GO f /Zll

0

Zl2

• ••

Xln\

0

•••

0 £ i i , #12, •• *, xin € £>

IV o

0 /

o

/ 2/n

2/12

0

• • •

0 \

2/21

2/22

0

•••

0

2/11, 2/21, • * • , 2/ml, 2/12, 2/22, ' ' ' , 2/m2 £ D

I

\2/ml

2/m2

0

•••

0/

(c)

/an

0

•••

0\

X21

0

•••

0 E n , #2i, • • • , x m i G D > ,

I V Zml

0

0/

3.3. Maximal Sets of Rank 2 f

/ 2/n

3/12

•

2/ln

2/21

J/22

■••

2/2n

0

0

•■•

0

[ \ 0

0

P

Q

103

2/11, 2/12, • • • , 2/ln, 2/21, J/22, • • • , 2/2n € I>

o/

(d) f / aril

0

a;2i

0

0\ 0 z n , ^2i, • • • , ^mi € D

l V zmi f

I

0

0/

/ 2/n

2/12

0

•••

0 \

2/21

2/22

0

•••

0

V2/ml

J/m2

0

where P € GLm(D)

••

Q | J/ll,

2/21, • • • , Vml, 2/12, 2/22, • • • , 2/m2 G -D

0/

and Q €

GLn{D).

Consider first Case (a). Clearly we can assume that Q = l(n\

Let

* — \PiJ h
Then the intersection is r

P

iin #21

^22

Zln\ Z2n

0

0

0

Io

2>12

P21


0

P22 Xu

X12

Xi n

.#21

X22

X2. n J

Pm2<

)=0

0 )

Let P21

P22


Pm2>

P2 =

Since P € GLm(D), P2 ^ 0. If P2 is of rank 1, then the intersection is the maximal set of rank 1 and, hence, the maximal set of rank 1 is contained in the maximal set of rank 2. If P2 is of rank 2, then the intersection is { 0 }. Case (d) can be treated similarly as Case (a).

Chapter 3. Geometry of Rectangular

104

Matrices

Now let us consider Case (b). Clearly we can assume that P = I^m\ the transformation

Then

X •—♦ XQ-1 leaves the maximal set of rank 1 invariant and carries the maximal set of rank 2 into f / 2/ii 2/21

2/12 2/22

0 0

••• •••

0\ 0 2/11, 2/21, ' * ' , 2/ml, 2/12, 2/22, * ' * , 2/m2 G -D

I V Vml Vml 0

••• 0)

Their intersection is f/x 0

y 0

0 0

l\0

0

0

••• •••

0\ 0

x,y € D

(3.16)

0/

For Case (c) we can prove similar to Case (b) that the transformation X P~XX carries the intersection to

(x

0

y 0 0 0 ^0

0

o\ 0 0

x,y € D

(3.17)

o7

We summarize the above discussion in the following proposition. P r o p o s i t i o n 3.18: Suppose that the intersection of a maximal set of rank 1 and a maximal set of rank 2 is nonempty. Then their intersection either consists of a single point, or coincides with the maximal set of rank 1, or is D a subset which is equivalent to (3.16) or (3.17) under Gmxn(D)> Definition 3.6: If the intersection of a maximal set of rank 1 and a maximal set of rank 2 is nonempty, does not consist of a single point, and does not coincide with the maximal set of rank 1, then it is called a plane. □ C o r o l l a r y 3.19: The parametric equation of a plane in MmXn{D)

is either

3.3. Maximal Sets of Rank 2

105

of the form (x 0

y 0

0 0

I \0

0

0

P

••• •••

0\ 0

Q + R x,y€D\

0/

or of the form (x 0 ••• 0 \ y 0 ••• 0 0 0 ••• 0 Q + R x,y € D 0 0

0/

where P € GLm(D), Q £ GLn(D), and i? G .M mxn (Z>). The parametric equation of a plane in the maximal set of rank 1 (3.4) is 3( 9l2> " ' " > 9ln) + J/(?21, 922, • • • , ?2n) + (<Jll, ^12) ' ' * , ^ln) x,y Q(m-l,n)

€Z?L

where ( • • • >
f / ( Pn \ P21

IV

x+

\Pml/

( Pl2 \ P22

\

/&ll\

y +

[/2i

Vfcml/

\Pm2/

Q(m,n-1)

x,y e D /

where ' ( p n , P21, • • •, Pmi) and '(pi2, P22, • • •, Pm2) are linearly independent.

From the discussion of Sections 3.2 and 3.3 we deduce Proposition 3.20: Denote the maximal set of rank 1 (3.9) and (3.10) by M. and Ai', respectively. Then the map

M (X11

£12

•••

0

0

•••

\ 0

0

—> AG'(n,

D)

Xln\

0

0 /

Q + R 1—► ( s n , £12, • • •, aji„)

Chapter 3. Geometry of Rectangular

106

Matrices

is bijective and carries lines and planes in A4 into lines and planes in AG\n, D), respectively, and similarly the map —* AGr{m,

M'

fyn

0 -" 0\

2/21

0

0

\ymi

o

0/

D)

(yn\ Q + R i-

2/21

\ym\)

is bijective and carries lines and planes in M! into lines and planes in AGr(m, D), respectively. Proof: The proposition follows immediately from Corollaries 3.13 and 3.19.

□ Therefore we may say that M. has an n-dimensional left affine space struc­ ture and M! has an m-dimensional right affine space structure.

3.4

Proof of the Fundamental Theorem

Now let us come to the proof of Theorem 3.4, i.e., the fundamental theorem of the geometry of rectangular matrices over any division ring. Proof of Theorem 3.4: We assume that D is a division ring and that both m and n are integers > 2. The second statement of the theorem is evident, so we prove only the first statement. Let ^ 4 b e a bijective map of AimXn(D) to itself and assume that both A and A"1 preserve the adjacency of pairs of points of Mmxn(D). By Corollary 3.6, A preserves the arithmetic distance between any pair of points of MmXn{D). Then A carries maximal sets of rank 1 into maximal sets of rank 1 and maximal sets of rank 2 into maximal sets of rank 2. We proceed in steps. (i) After subjecting A to the bijective map of the form (3.1) Xv-^X-

A{V)

for all X G

Mmxn(D),

we can assume that A(0) = 0.

(3.18)

3.4. Proof of the Fundamental Theorem

107

(ii) Let Mi be defined by (3.4). We know that Mi is a maximal set of rank 1, hence A(Mi) is also a maximal set of rank 1. Since 0 G A
2/12

I/ln\

0

0

o

Vo

o

o )

/ J/ii J/21 \ym\ and Q

0 0 0 yii, y2i, •••, j/mi € z? 0 0 € GL„(D). After subjecting .4 to the bijective

(V\\

A{Mi)

or A{Mi)

f = <

where P € GLm(D) map X - ^ P-^Q"1

Q S/ii, 2/12, ••, Vin € D

for all X 6 .M m x „(I>),

we can assume that A(M1)

= Mi

(3.19)

A(Mi)

= M[,

(3.20)

or where A^j is defined by (3.5). Notice that A carries lines and planes in M.\ into lines and planes in A{M.\), respectively. By Proposition 3.20 the bijective map AG'{n, D)

Mi I X\i X12 0 0

Vo

o

Xln\

0

•••

(xn,a;i2,---^in)

o /

carries lines and planes in Mi into lines and planes in AGl(n, D), respec­ tively, and the bijective map 0\ 0

AGT(m, D) (Vn\ V2l

0/

\ymij

M\ /j/ii yai

0 0

\ymi

0

••• •••

Chapter 3. Geometry of Rectangular

108

Matrices

carries lines and planes in M'x into lines and planes in AGr(m, D), respec­ tively. If (3.19) occurs, then A induces a bijective map from AGl(n, D) to itself, which carries lines into lines and planes into planes. Suppose that I' x\\

0

0

0

0 )

0

V 0

x

12

0

0

0

\o

0

0 /

then by Theorem 2.7 (the fundamental theorem of the affine geometry over any division ring), we can assume that (Xll>

X

12> • • • '

X

ln)

=

(XlU

X

™1 •"» 3 > l n ) T + ( a i , 0-1, " • , «„),

where a is an automorphism of D, T € GLn(D), and (ai, a 2 , • • •, an) € Z)(n). Since A(0) = 0, we have (a\, a 2 , • • •, a n ) = 0. Thus after subjecting A to the bijective map X .—► (XT-1)"'1

for all X € A4mx„(I>)>

which is of the form (3.2), we have x

( X\\

\n\

0

0 0

x

\n\

0 0

0

X\i

0 0

(3.21)

0 / \ 0 0 for all i n , £i25 • • •, £in € D. That is, .4 leaves every element of Mi in­ variant. If (3.20) occurs, then A induces a bijective map from AG'(n, D) to AGr(m, D), which carries lines into lines and planes into planes. By Theorem 2.11, we have m — n. Suppose that / xu 0

x12 0

V 0

0

0 )

0

/yn Vn

0 0

0 /

Wm 0

0\ 0

then we have *(yn, 2/2i, • • •, 2/ni) = Ttftxn,

x12, • • •, xi n ) T ] + *(ai, a 2 , • • •, a n ),

3.4. Proof of the Fundamental Theorem

109

where r is an anti-automorphism of D, T € GLn(D), and (ai, a 2 , • • •, a n ) G Z)(n). Since A(0) = 0, we have (ai, a 2 , • • •, a n ) = 0. Thus after subjecting A to the bijective map X _+ '((T-^r1)

"for all X € M»x»(I>).

which is of the form (3.3), we also have (3.21). Therefore we can always assume that (3.21) holds. (iii) Clearly, both Mi and M[ are maximal sets of rank 1 containing 0 and En. By Corollary 3.10 there are two and only two maximal sets of rank 1 containing two given adjacent points. Since A(Mi) — Mi, we also have A(M'i) = M\. Suppose that (Vxi 2/2i \ym\

0

0

0

0

0

0/

o/

(3.22)

then by the fundamental theorem of the affine geometry we have

(yiAT1 2/21

= P

Uml/ where P € GLm(D)

2/21

\ymi/

and rx is an automorphism of D. Let P

=

\Pij )l
Substituting xEu into (3.21) and (3.22) and equating the results, we obtain

/xn\ 0 V0 /

= P

0

Vo/

Substituting i n = 1 into the above equation, we obtain Pll = 1, P21 = P31 = • * * = Pml = 0.

Chapter 3. Geometry of Rectangular

110

Matrices

Thus a n = Xi\ for all Xu £ D, hence Ti is the identity automorphism of D, i.e., ri = 1. After subjecting A to the bijective map X i—► P-*X

for all X 6

Mmxn(D),

which leaves every element of M.\ invariant, besides (3.21) we also have

(Vu

0 •

Jf21

A

0

\ymi

• °) 0

• ■

0 •

=

• oj

(yii S/21

0 0

• •

• 0\ • 0

\ J/ml

0

•

• oj

for all yn, yn, ■• •, Vmi € I>. (3.23)

(iv; Let / 0

0

0

0

0 \

M= <

x

x

ili

0

i2i

' ' ' j xin £ A/

, i = 1, 2, •••, m,

0

IV o o

0 )

and

M'j=

f/0 0 {

0 0

yy V2j

0 0

0\ 0 yij> 3/2J, • • • , ymj; G -D

0

ymj

0 j = 1, 2, •••, n.

A ^ is a maximal set of rank 1 containing both 0 and En, which are left fixed by A according to (3.23). Then by Corollary 3.10, A{Mi) = Mi, i = 1, 2, • • •, n. M'j is a maximal set of rank 1 containing both 0 and E\j, also by Corollary 3.10, A(M'j) = M\, j = 1, 2, • • •, m. Assume that / 0

0

•••

0 \ 0

/ 0

0

0

Ijl

X,2

^•l

0

0

0

Vo o

o /

0

0 0

•••

0 \ 0

x

i2

0

Vo o

oJ

3.4. Proof of the Fundamental Theorem

111

and /O 0

... •••

Vo

0 0 0

0 0

0\ 0

/0 0

0

yli

y2j

o

y% o

ymj

0

0/

Vo

o sC o

Vlj

0

o\ 0

then by the fundamental theorem of the affine geometry we can assume 0 / that « 1 > <2> * * • i X*in) =

fa,!,

&t2, ' * ' , S t n J ^ Q t ,

t = 1, 2 , • ■ • , m ,

(3.24)

and

fyn\'J , i = i,2,

= Pj

(3.25)

\VmjJ

\VmjJ

where Qx = /(»>, Q, € G£n(£>) (* = 2, 3, ■ • •, m), P x = 7<m>, P, € G£m(/J>) (j = 2, 3, ••-, n), o-! = Ti = 1, and }, we must have A(xijEij) = x'jEij,

I
1 < j < n.

It follows that both Qi {i — 2, 3, • • •, m) and P, (j = 2, 3, • • •, n) are diagonal matrices. Let \

qn Qi =


\

/Pij

and

P2j

Pj = \

tin/

Pmj/

Comparing (3.24) and (3.25), we obtain XijQij = PiJX?ji

for

a11

X

ij ^ £>•

Substituting x t j = 1 into the above equation, we obtain q^ = p t J . But when i = 1, we have cri = 1 and qij = 1 for all j = 1, 2, • • •, n, thus r,- = 1 and P!j = 1 for all j = 1, 2, • • •, m. Similarly, when j = 1, we deduce crt- = 1 and

112

Chapter 3. Geometry of Rectangular

qn = 1 for all i = 1, 2,

m. For any (#,i, £t-2, • • •, x%n) ^ 0, the points / 0

0

0

\ 0

0

0

and

0 \ 0

0

0

Xn

Xi2

0

0

o

^ 0

0

o I

0 /

•

0

Matrices

are adjacent, thus their images under A / 0 I X{1

0 \ 0

Xi2

•■•

0

■••

0

X{n \

0

and

0 /

0

0

0

Xil

Xi2

0

0

Vo o

o\ Qi

o )

are also adjacent. Hence (x,i, ^2? *•? #»n) and (x t i, x,2, •••, %in)Qi are linearly independent, i.e., (# t *l, X t 2, * * '

7

%in)Qi

=

^i\xili

x

i2i

' * ' j #tn)

for some //» € .D*. Substituting a\i = X{2 = • • • = £m = 1 into the above equation, we obtain qn = qi2 = — • = qin = /zt-. But #,1 = 1, therefore Qi = /("> for alH = 1, 2, • • •, ra. Similarly, P, = /<m) for all j = 1, 2, • • •, n. Therefore •4(X) = X

for all X € M i , M2, ■ • •, -Mm, and A
(v) We prove that A{X) = X for all X G Mmxn(D). We need to distin­ guish the cases m < n and m > n. Since they can be treated in a similar way, we consider only the case m < n. Thus in the following we assume that m < n. Consider first the case when X is of rank ra. Let A(X) — X*. Since A preserves the arithmetic distance between any two points of A4 mXn (Z)) and

3.4. Proof of the Fundamental Theorem

113

A(Q) = 0, X* is also of rank m. Write

x=

and

X* =

*

x

2 5

where a?i, x2, • • • > ^m? #i? £2> '"' > ^m a r e ^-dimensional row vectors over D. By (3.26), for any A2, • • •, Am G D A2x2

lX\x -— A 2^2 £2 (x

Am^m \

A7711,3^771 1

4 \

*•— * AA x m\ \ mX mm 0

0

0

—

0 •••

/X! (x\-

y

0

1

Since - Amxm \

/ x i - A2x2

/ A2z2 H

0

X -

I

=

0

\

)

h Amxm ^ x2 xm

/

is of rank m — 1, /xi

X2x2

X*\

( xx ■-

Amxm\

0

xi + \2x2 + -

1

^ 7 7 1 *E 771 ^

x

2

—

0

)

\

X

m

/

is also of rank m — 1 for all A2, • • •, Am G i}. Thus m

Xi=x{

+ E Mj*jN i=2 771

£fc = E /^fcj^j, fc = 2, 3, •• £fc= • ••,•, m, m, where /Zj (j = 2, 3, • • •, m) and fikj (&, i = 2, 3, • • •, m) are elements of D. Applying the same argument to the z'-th row (i = 2, 3, • • •, m), we obtain

*,-=*; +z; A*}0*;, • • ••,• m, Xk*== EE /44 H j ^fcfc == i,1, 2, •, m, « 2,..• • ,•,* , 2,

Chapter 3. Geometry of Rectangular

114

Matrices

where i denotes that the index i is deleted, and fij (j = 1, 2, • • •, z, • • •, m), rkj (A:, j = 1, 2, • • •, i, • • *, m) are in D. Eliminating Xi, we obtain \~^

*

(t)

m X~^

*

*

Since rankX* = m, a;J, a^, • • •, x*m are linearly independent. Thus \ii = 0,

i = 2, 3, • • •, m.

Therefore xj = a?i. Similarly a:* = #; (i = 2, 3, • • •, m). Hence A(X) when X is of rank ra.

= X

(vi) Now consider the case when X is of rank < ra. We distinguish the following two cases. (a)

D ± F 2 . Let A(X) / #11

#12

= X* and write * ' '

( xxx

Zin \

X ="

, x* = \ £ml

£ra2

"* *

^mn '

x12

•^21

\X*ml

*ln\

^22

<

^ n

/

If Zn ^ 0, then by Step (v), for any A2, • ■ •, Am ^ 0, we have /xn '

x12 A2

■•■ xlm

V

xi i m + i

■

/

A„ fxn

£in\

X12

•••

Zim

xi,m+i

Zln\

A2

Since X and I xn

V

x\2 A2

•••

xim

zi,m+i

•••

£in\ (3.27)

3.4. Proof of the Fundamental

Theorem

115

are of arithmetic distance < ra, X* and (3.27) are also of arithmetic distance < ra, i.e., //xl * x?- ! - *xn „ X

V

21

X X*12-X 12 \2~ ~M

x

X

Tl~


X

lm

^2

X

* X

lm

x

l

m -|-l X

2m

X

X

X\n

X\n X

X

mm ~ *rn

m2

#l,m+l

2,m+1

X

m,m+l

\

2n

mn

1

is of rank < ra for any A2, • • •, Am ^ 0. Since D =^ F 2 , we must have #ii = x\\. If x ^ ^ 0, then considering *4 _ 1 , we obtain also x ^ = x\\. Therefore we always have x\x = i n . Similarly, we can prove that x*- = xtJfor all i = 1, 2, • • •, m; j = 1, 2, • • •, n. Therefore we also have A(X) = X when X is of rank < ra. (b)

D = F 2 . Consider first the case m = n = 2. We already have

and for for X X having having rank rank 2. A{X)-for X X e£M •A(A") ==XX for Mx,i,MM2,2,M\ M[, ,M' M'22 and 2. If X =---11and M[, A M^2,, then then X X must must be andXX ^<£ Mi,.M M22, , -Mi, If rank rankX .Mi, be X

-c

!)•

Therefore ! ) ■ ■ ( !

"(!

!

)

■

Hence we have A(X) = «4(*) =-XX

for all X X € M M22(D). (D).

Then consider the case n > 3. Let -4(X) = X* and write

X =

(Xlx\

2

, x* =

x

2 ?

\xm/ where Xi, x 2 , • • •, x m , x^, x^, • • •, x ^ are n-dimensional row vectors in 1^2 . Assume that X is of rank ra — 1. Then there must be m — 1 linearly in­ dependent vectors among xi, x 2 , • • •, xm. Assume that x 2 , x 3 , • • •, xm are

Chapter 3. Geometry of Rectangular

116

linearly independent. There are 2 n — 2 m such that

I Ax Ast

x=

/ Aj

1

Matrices

choices (Ai, A2, • • •, An) € F?

* * *

K\

A2 ^ 2 • • • AB \

X=\

\

X-m

)

\

Xm

/

is of rank m. For any such choice, by Step (v) is of rank m. For any such choice, by Step (v)

A(X)-- = X.

Let x\

— \XW>

A(X) = X. • • •, xj n ). Since X and X are adjacent, the matrix

X

12i

Let x\ = (xln x{2i "',# 1 1 ^in)* X^2A2 and*•X are adjacent, the matrix x# 11 22 —— *• *• *I»--A„ # i n — ^n \ \ ( *n -~~A*1! Since x2 — Xi

\

X

m ~

X

m

(3.28)

/

is of rank 1. If x2 =fi X2, then there are only two choices of (Ai, A2, •••, ^n) € F2 such that the first two rows of (3.28) are linearly dependent. Since n > m and n > 3, we have 2 n — 2 m _ 1 > 4. Thus there is a choice of (Ai, A2, • • •, An) G F2 s u c n that X is of rank m and the first two rows of (3.28) are linearly independent, which is a contradiction. Therefore we must have x2 = x2. Similarly, we have #* = X{ for i = 3, 4, • • •, m. Thus (*i\

A(X) =

X2 X2

\xm) If xi 7^ 0, then there exist indices z2, Z3, • • •, im-i with 1 < i2 < ^3 < • • • < im-i < m such that xi, x,-2, zt-3, • • •, X{m_r are linearly independent. By the above argument we can prove that x\ = x\. If x\ = 0, then we must have x\ = 0, for otherwise, considering A'1 will lead to Xi =£ 0, which is a contradiction. Hence A(X) ===XX for all X of rank m —1 .1. A(X) m -Now assume that X is of rank m — 2. We may assume that £3, £4, • • •, x rn are linearly independent. There are 2 n — 2 m 2 choices of (Ai, A2, • • •,

3.4. Proof of the Fundamental

Theorem

117

An) € F(2n) such that / Ai

A2

...

An \

Jf = \

*m

/

is of rank m — 1. Then by the above discussion we have A(X) = X. Since X and X are adjacent, / x*xl - Ai

z j 2 - A2

I

• • • x\n - \n \

#2 ~~ ^2 £ 3 — a: 3

I

is of rank 1. As in the case when rank X = m — 1 we can prove that x* = xt(i = 2, 3, * • •, m). Also similar to the case when rankX = m — 1, we can prove that x\ = x\. Hence A(X) = X

for all X of rank m — 2.

Proceeding in this way, we obtain A(X) = X

for all X G A4™xn(F2).

□ Without essential difBculty, Theorem 3.4 can be generalized as follows. Theorem 3.21: Let D and D' be division rings, m, n, m', and n' be integers > 2, Mmxn(D) be the space of m x n matrices over D, and jVfm/xn/(Z)') be the space of m' x n' matrices over D'. Let A : M.mXn(D) *-> -Mm'xn'(Df) be a bijective map and assume that both A and A-1 preserve the adjacency of pairs of points. Then m = ra;, n = n' or m = n', n = m'. In the first case, D is isomorphic to D1 and .4 is of the form A(X)

= PX°Q + R

for all X G A4mxn(#),

(3.29)

118

Chapter 3. Geometry of Rectangular

Matrices

where a is an isomorphism from D to D'y P G GLm/(Df), Q G GLnt(D')y and R € M.m'xn'(D'). In the second case, D is anti-isomorphic to D' and A is of the form A(X) = P\XT)Q

+R

for all X G Mmxn{D),

(3.30)

where r is an anti-isomorphism from D to I?', P G GLm>(D'), Q G GLn>(D'), and i? G -M m / X n'(^')- Conversely, any map from MmXn{D) to M m / X n / of the form (3.29), when m = m' and n = n', is bijective and both the map and its inverse preserve the adjacency of pairs of points, so is any map from •Mmxn to A4m'xn' oi the form (3.30), when m = n' and n = m'. □

3.5

Application to Algebra

It is well-known that the set o f n x n matrices over a division ring D, A4n(D), is a simple associative ring with respect to the addition and multiplication of matrices and is called the total matrix ring of degree n over D. As an application of Theorem 3.21 we have Theorem 3.22: Let D and D' be division rings, n and n' be integers > 1. If there is an isomorphism from the total matrix ring Ain(D) to Aini(D'), then n = n' and D is isomorphic to D' or anti-isomorphic to D'. In the first case, the isomorphism is of the form X h-> Q~lX°Q, where a is an isomorphism from D to D' and Q G GLn(D'). case, the isomorphism is of the form X^Q-xt(XT)Q,

(3.31) In the second

(3.32)

where r is an anti-isomorphism from D to D'. Proof: Let A be an isomorphism from_M n (D) to Mn'(D'). If n = 1, then Mi(D) = D has no zero divisors. However, if n' > 1, then Mn'(D') is a ring with zero divisors. Therefore, when n = 1 we must have n' = 1 and A is an isomorphism from D to D'. Hence our theorem is true for n = 1.

3.5. Application to Algebra

119

Now assume that n > 1, then we also have n' > 1. Since *4(X + Y) = A(X) + A{Y) for all X, Y G X n (Z>), if we show that both A and ^ l " 1 carry matrices of rank 1 to matrices of rank 1, then both A and A~x preserve the adjacency of pairs of n x n matrices, and Theorem 3.21 can be applied. A nonzero element A G M.n(D) is called an idempotent matrix if A2 = A. Two elements A and B of Mn(D) are said to be orthogonal if AB = BA = 0. An idempotent matrix A G A4n(.D) is called primitive, if it cannot be written as a sum of two orthogonal idempotent matrices. It is well-known that an idempotent matrix is primitive if and only if it is of rank 1. All J5n, £ 2 2, * * *5 Enn are primitive idempotent matrices. It follows from the above definitions that the isomorphism A carries primitive idempotent matrices into primitive idempotent matrices. Furthermore, we have the following characterization of matrices of rank 1. L e m m a 3.23: A nonzero n x n matrix M is of rank 1 if and only if there is a primitive idempotent matrix E such that EM = M, and if and only if there is a primitive idempotent matrix E' such that ME' = M. Proof: Let M be a nonzero n x n matrix of rank 1. By Proposition 1.17 there exist P , Q G GLn(D) such that PMQ = Let E = P~xEnP,

Eu.

then E is a primitive idempotent matrix and

EM = P~1E11PM Similarly, let E' = QEuQ'1, ME1 = M.

= P^EnEuQ'1

= P^EuQ'1

= M.

then E' is a primitive idempotent matrix and

Conversely, if E is a primitive idempotent matrix such that EM = M . By Proposition 1.22, r a n k M < ranki? = 1. Since M ^ 0, M is of rank 1.

Chapter 3. Geometry of Rectangular

120

Matrices

Similarly, if E' is a primitive idempotent matrix such that ME' = M , then n

M is also of rank 1.

Proof of Theorem 3.22 (continued): Lemma 3.23 implies that both A and A'1 carry matrices of rank 1 to matrices of rank 1. Consequently, both A and A'1 preserve the adjacency of pairs of matrices. By Theorem 3.21, n = n', and D is isomorphic to D' or anti-isomorphic to D'. In the first case, A is of the form A(X) = PXaQ

+ R

for all X G

Mn{D),

where a is an isomorphism from D to Df, P , Q G GLn(Df), and R G Mn{D'). But .4(0) = 0, therefore R = 0. We also have A{I) = J, therefore PQ = I. Hence A is of the form (3.31). For the second case we can prove in a similar way that A is of the form (3.32). □ Definition 3.7: Let R and R' be rings with identity element. A bijective map a: R —► R' a i—> a a , is called a semi-isomorphism conditions hold.

from R to R' if for any a, b G R the following

(a + by^a'

+ V,

(abay = aabaaa, l a = l. D

Set 6 = 1 into the second equation and use the last equation, we deduce

(«2r = K) 2 Clearly, isomorphisms and anti-isomorphisms are semi-isomorphisms. For the total matrix rings Mn(D) and Mn(D') over division rings D and D\ respectively, we have Theorem 3.24: Every semi-isomorphism of the total matrix ring Mn(D) to Mn'{D') is either an isomorphism or an anti-isomorphism. More precisely, if A is a semi-isomorphism from Mn(D) to Mnt(D'), then n = n' and D

3.5. Application to Algebra

121

is isomorphic to D' or anti-isomorphic to D'. In the first case, the semiisomorphism is an isomorphism and is of the form (3.31) X >—►

Q^X'Q,

where a is an isomorphism from D to D' and Q £ GLn(Df). In the second case, the semi-isomorphism is an anti-isomorphism and is of the form (3.32) X_-Q-"(JT)Q, where r is an anti-isomorphism from D to D'. Proof: Let A be a semi-isomorphism from Mn(D) to Mn'(D'). If n = 1, then n' = 1 and our theorem is just Theorem 2.25. Now assume that n > 2, then we also have n* > 2. For any X , Y € A4n(2}) we have A(X + Y) A(XYX) A(I) A(X2)

= A(X) + A(Y), = A(X)A(Y)A(X), = I, = A(X)2.

Therefore A(0) — 0, A(I) = / , and A carries idempotent matrices into idempotent matrices. Moreover, we have A(XY

+ YX) = A((X + Yf -X2 - Y2) = (A(X) + A(Y)f - A(X)2 = A(X)A(Y) + A(Y)A(X).

A(Y)2

Let X be an idempotent matrix and XY = YX = 0. Then 0 = A(XY

+ YX) = A(X)A(Y)

+

A{Y)A(X).

It follows that 0 = A(XYX)

= A(X)A(Y)A{X)

= -A(X)2A(Y)

=

-A(X)A(Y).

Therefore A(X)A(Y)

= A(Y)A(X)

= 0.

Thus A carries orthogonal idempotent matrices into orthogonal idempo­ tent matrices. Hence A carries primitive idempotent matrices into primitive

122

Chapter 3. Geometry of Rectangular

Matrices

idempotent matrices. Then our theorem will follow from Theorem 3.21 and the following lemma. L e m m a 3.25: A nonzero n x n matrix M is of rank 1 if and only if there exists a primitive idempotent matrix E such that (/ - E)M(I

-E)

=0

(3.33)

and (M - EMEf

= 0.

(3.34)

Proof: Let M be a matrix of rank 1, then there exist P , Q G GLn(D) that PMQ = Eu.

such

Let E = P - ^ n P , then we can verify that both (3.33) and (3.34) hold. Conversely, without loss of generality, we take E = En. we have

Then from (3.33)

* - ( : !)• where b is an (n — l)-dimensional row vector and c is an (n — l)-dimensional column vector. From (3.34) we have

* ) 2 = 0,

(M-EMEf={°c

then we deduce either b = 0 or c = 0. Therefore M is of rank 1.

□

Now assume that D is of characteristic not two. Then it is also known that Mn{D) is a Jordan ring with respect to the matrix addition and the Jordan multiplication (or the symmetrized multiplication) X-Y

=\(XY

Denote this Jordan ring by J(A4n(D)). A:

J(Mn(D))

+

YX).

A bijective map —►

J{Mn,{D'))

is called an isomorphism of Jordan rings, or a Jordan isomorphism, if for any X, Y e J(Mn(D)) A{X + Y) = A(X) + A(Y),

(3.35)

3.6. Application to Geometry

123

and A{X • Y) = A{X) • A(Y).

(3.36)

L e m m a 3.26: Assume that both D and D' are of characteristic not two. Then every isomorphism of Jordan rings J(Mn(D)) and J(Mni(D')) is a semi-isomorphism of the total matrix rings Mn(D) and Mni(D'). Proof: Let A be a Jordan isomorphism from J(Mn(D)) is easy to verify the following identity 2(XYX)

= {XY + YX)X

+ X{XY

to J(Mn'(D')).

+ YX) - {X2Y +

It

YX2).

From (3.35), (3.36), and the above identity we deduce A(XYX)

=

A(X)A(Y)A(X).

Let X = I into the above identity, we obtain A(Y) = A(I)A(Y)A(I). Since A is bijective, A(I) must be invertible. Substituting X = Y = / in (3.36), we obtain A(I) = A(I)2. Therefore A(I) = / . Hence A is a semi-isomorphism.

a From Lemma 3.26 and Theorem 3.24 we deduce immediately Theorem 3.27: Let D and D' be division rings both of characteristic not two and n and n' be integers > 1. Then every isomorphism of Jordan rings J(Mn(D)) and J(Mn'(D')) is either an isomorphism or an anti-isomorphism of the total matrix rings Mn(D) and Mnf(Df). □

3.6

Application to Geometry

Let D be a division ring, m and n be integers > 2, and PGl(m + n — 1, D) be the (m + n — l)-dimensional left projective space over D. The set of (m — l)-flats in PGl(m + n — 1, D) is called the left Grassmann space of (m - l)-flats in PGl(m + n - 1, D) and denoted by Glm+n-i}m-i(D)> a n d the (m — l)-flats are called the points of ^ + n - i , m - i ( ^ ) - ^ n ( m — 1)-Aat W corresponds to an m-dimensional subspace of the (m + n)-dimensional row vector space over Dy which will also be denoted by W, and the

124

Chapter 3. Geometry of Rectangular

Matrices

m-dimensional subspace W can be represented by an ra x (ra + n) matrix of rank ra over £>, which is also denoted by W. We say that the ra x (ra + n) matrix W of rank ra is a homogeneous coordinate of the (ra — l)-flat W. Two ra x (m + n) matrices W and W\ both of rank ra, are homogeneous coordinates of the same (ra — l)-flat if and only if there is a P £ GLm(D) such that W — PW. Let us write the homogeneous coordinate W of an (ra — l)-flat W in the block form

W = (X

Y),

where X is an ra x n matrix and Y an ra x ra matrix. If rank F = ra, then

( y - 1 * /) is also a homogeneous coordinate of W, and we call W a /mz£e pom£ of Glm+n-i m -i(-D) a n d t n e raxn matrix Y~1X the non-homogeneous coordinate of VF. If r a n k F < ra, then W^ is called a point at infinity. Thus we also call Qlm+n_lim_1(D) the protective space of ra x (ra + n) matrices over Z? and denoted it by P M m x ( m + n ) ( ^ ) ' The elements of VMmx(m+n)(D) are called the points. PGLm+n(D) can be regarded as a group of motions on > VM mx(m+n)(-D )i that is, each element T £ PGLm^.n(^D^ defines a bijective map from /PA^mx(m+n)(^D) to itself W »—> WT.

(3.37)

Proposition 1.24 implies the following proposition. Proposition 3.28: The group PGLm+n(D)

acts transitively on the space

VMmx(m+n)(D).

D

Parallel to Definition 3.1 we have Definition 3.8: Two points W\ and W2 of VMmX(m+n)(D) are said to be of arithmetic distance r, denoted by ad(Wi, W 2 ), if W1UW2 is an ( r a + r — l)-flat or (Wi\

ran

HvK 2 J = m + r '

where W\ and W2 are matrix representations of the points W\ and W2, respectively. If r = 1, then W\ and W2 are said to be adjacent (or coherent). D

3.6. Application to Geometry

125

It is clear that the map (3.37) preserves the arithmetic distance and, in particular, the adjacency. Our purpose is to characterize the bijective maps from VMmx(<m+n)(D) to itself such that both the maps and their inverse maps preserve the adjacency of pairs of points. Parallel to Proposition 3.2 we have Proposition 3.29: Let Wu W2, W3 G VMmX(m+n)(D).

Then

W22)) > 0; ad(Wi, W 1° ad(Wi, W W22)) = 0 if and only if Wx1 === W w22. . 2° ad(Wi, W W22)) = ad(W 22, Wi).

3° ad(Wi, W22)) + ad(W 22, W W33)) > ad(Wi, W W33). Proof: 1° and 2° are evident; 3° is left as an exercise.

□

Proposition 3.30: Let W\ and W2 be two points in VMmx(m+n)(D), which are of arithmetic distance r. Then they can be carried simultaneously to

(0/<») and ( ( 7* ' °) (0 j(**h and «

respectively, under the group

:

/(/<-,), ••)■

PGLm+n(D).

Proof: By Proposition 3.28 we can assume that m

) W1 === (0 (0 /(J ( m)). ). Wt

Write W2 in the block form

w22--== (x (X2 rYi), 2 ), where X2 is m x n and Y2 is m x m. Since ad(Wi, W2) = r, we have rankX 2 = r. By Proposition 1.17 there is an element P G GLm(D) and an element Q G GLn(D) such that

«*-(? V o 2). PX2Q == Then the transformation

^W(WQ w^ W ■—>

(^

m )) ) // (( m

for all W W€

VM VMmx{m+n) (D) mx{m+n)(D)

126

Chapter 3. Geometry of Rectangular

Matrices

leaves W\ invariant and carries W2 to

1

-a :

w3 =

«).

oj

If I3 is invertible, then the transformation

M w*» ~ M' » ' ( r

i

n-)

W

«■■)

leaves W\ invariant and carries W3 to

((T OJ '">)• ((7 I) /(•»)),

and the proposition is proved in this case. If Y3 is not invertible, we write

HlW' *-(£)• Y3 =

where I31 i s r x m and I32 is (m — r) x m, then Y32 is of rank m — r. There is an r x m matrix B such that (3.38)

( % : * ) is invertible. Then the transformation fj{r) W i - -► w\

B

j

V{

/(n-r)

\

/<•»>/

leaves Wi invariant and carries W3 to

-((7

w4 =

n),

where 5^ is (3.38), which is reduced to the above case.

□

Proposition 3.31: The elements of the group PGLm+n(D) leave the arith­ metic distance between any two points of VAimx(m+n){D) invariant and for every r with 1 < r < min{m, n } , the set of pairs of points of arithmetic distance r in VMmx(m+n)(D) form an orbit under the group PGLm+n(D). D

Parallel to Definition 3.2, Proposition 3.5, and Corollary 3.6 we have

3.6. Application to Geometry

127

Definition 3.9: Let Wu W2 G VMmx{m+n)(D). When Wt ^ W2, they are said to be of distance r, denoted by d(Wi, W2) = r, if r is the least positive integer for which there is a sequence of r + 1 points W^°\ W^\ • • •, W^ with W^ = Wi and W^ = W2 such that W® and W^* 1 ) are adjacent, i = 0, 1, 2, • • •, r - 1. We define also d(W, W) = 0. □ Proposition 3.32: For any two points Wi, W2 £

VMmx(m+n)(D)

ad(Wi, Wi) = d(Wi, W 2 ). D

Corollary 3.33: Let ^4 be a bijective map from VMmx(m+n){D) to itself such that both A and »4 _1 preserve the adjacency of pairs of points in / PA1 rnX (m+n)(^) ) then A also preserves the arithmetic distance between any pair of points in VMmx(m+n)(D). □ Definition 3.10: Let A4 be a non-empty set of points in 'P./WTnX(m+n)(Z}). M is said to be a maximal set of rank 1, if any two points of M are adjacent and there is no other point outside M., which is adjacent to each point of M. D Proposition 3.34: Any maximal set of rank 1 in VMmx{m+n)(D) is carried into a maximal set of rank 1 under any transformation of the form (3.37). O Proposition 3.35: Both the set ( / aZm ll 0 >f 0 Mx = \I 1| :

X12 Xi2 0 0 :

0 0 ••• 0 \ I 1 1 1 * n , - - -•,^i,n+i , s i f n + i€€ #0 and andl1 ,,, aj .. I (^ii5 » i,n+i) ^ 0 ( z n r ••»«l n+l) 7^ I[ f

•••• Xi |n+ i 0 •••• 0 •••• •

0 •••• Iv I I 00 0

00

i1 /) \ *>-

v

JJ

'

rw_1

m-1

and the set

and the set / f // xZul l £21 A < 2 =
0 ••• 0 1 0 ••• 0 1 0 ••• 0 1

0 ••• 0 . .

\

1

0 ••• 0 n-1

a?ii,^2i,* •

.

I lUml 0 ••• 0 \ ^ml

V

\ I \\xlux2U'",xmle

11/1 / rn

] •, x m J G Z? >

D>

JJ

Chapter 3. Geometry of Rectangular

128 0 0 0

0 0 1 1

0

0 0 0 0 0 0 0 0

0 0

0 0

0

0

[ \ 0 0 ••• 0

0 0

0

0

f/i

uu i=2

o

0 0

x

0

3,n+i

1

t—1

n

0 %2,n+i

XitH+i

Matrices

0\ 0 0

0

1

1

1/ m—t

are maximal sets of rank 1. Moreover, every maximal set of rank 1 can be carried under a transformation of the form (3.37) to either M\ or M.2Proof: We prove only the second statement of the theorem. Let M. be a maximal set of rank 1 in VMmX(m+n)(D). By Proposition 3.30 we can assume that A4 contains the following pair of adjacent points Wx = (0 J ( m ) )

and

J< m) ),

W2 = (En

where 0 and En are both m x n matrices. Let W = (X F ) , where X is mxn and Y m x m, be another point of M. Since W and W\ are adjacent, X is of rank 1. Thus r a n k y > m — 1. We distinguish the following two cases. (a) r a n k F = m. We can assume that Y = I^m\ Since W and W2 are adjacent, and (W2\_

(En

I \ _ ( I

0W

\WJ

U

IJ~\I

IJ\X-En

then W = (X

En

1^).

I\

0J'

X — En is of rank 1. From r a n k X = ra,nk(X — Eu) = 1, we deduce that W is either of the form 1 0

0

V 0

0

\

0 (3.39) 1/

3.6. Application to Geometry

129

or of the form x2i

0

\ xm\

0

0 0

1 1 (3.40)

(b)

1/

r a n k F — m - 1. By Proposition 1.23 Y has an echelon normal form (I

0

0-

0

0-

V0 0 •

X\,n+i X2,n+i

0 ••• 0 \ 0 ••• 0

1 0

3i-l,n+i 0

0 ••• 0 1

0 0

0 0

for some i (1 < i < m).

1 0

Then we can assume that 0

( 0 0 1 1 Y = 0 0

1 0

Vo o

0

o\

0 0 Xl,n+i 0 Xz,n+i 0

0 0 for some i (1 < i < m).

Xi,n+i 0 0 1 0 1

t'-l

When i = 1, we can assume that W is of the following form /Xn

X-12

0 0

0 0

V 0

0

xm 0 0

0 1 1

(3.41) 1/

Chapter 3. Geometry of Rectangular

130

Matrices

where (^n, £i 2 , • • •, Xin) =fi 0, and when i > 1, we can assume that W is of the following form

(1 0 • • 0 0 0 • • 0 0 0 • • 0 1 0 0 • • 0 1 0 0 • • 0 0 0 • • 0 \0

0

0

0\

%3,n+i

for some i (1 < i < m).

1

£i,n+i

1

0

0

•

Z2,n+t

1/

(3.42) Clearly, any two points of the form (3.39) or (3.41) are adjacent, any two points of the form (3.40) or (3.42) are adjacent, and a point of the form (3.39) or (3.41) is not adjacent with a point of the form (3.40) or (3.42). a Therefore M is carried to either A^i or A<2. Corollary 3.36: Given any pair of adjacent points in VM.mx{m+n(D), there are two and only two maximal sets of rank 1 containing both of them. □

Denote the intersection of M\ and A42 by £, i.e., •

c= \

(x 0 ••• 0 y 1 0 0 ■•• 0 0 0 ••• 0 \0

0 ••• 0

\ 1 1

x,yeD \x, ye

and (x, y) ^ (0, 0)

1/

Corollary 3.37: The intersection of two maximal sets of rank 1 which contain more than one point in common can be carried into C under a transformation of the form (3.37). □ Definition 3.11: If the intersection of two maximal sets of rank 1 contains more than one point in common, then it is called a line. □ Corollary 3.38: There is one and only one line passing through any pair of adjacent points. □ Proposition 3.39: The following map from the maximal set of rank 1 A41

3.6. Application to Geometry to PGl(n,

131

D) X\2

■

0 0

• •

0 0

0 0 1 1

0 0

Vo o

o\ ( z n , ^12, • • •, ^l.n+i) (3.43)

o

i/

is bijective and carries lines in M\ into lines in PGl(n, D). The following map from the maximal set of rank 1 M2 to PGT(m, D) ( su

0

£21

0

V Xml

0

/ 1 \

\

0 0

x21 1/

(3.44)

\xmlJ

n-1

/1

0 ••

0 0

00-

0 0 0

0 0

00-

0 0

0

0-

0

0

0 0 1 1

1

0

0

£2,n+i

0

Z3,n+t'

0

/

0\ 0 0

3t,n+t

-1 0

0 0 0

0

0

\

X3,n+i

Xi,n+i 0

0

0 X2,n+i

1

0 0

V 0

) (3.45)

r

is bijective and carries lines in M.2 into lines in PG (m,

D).

Proof: By Corollary 3.37, a line can be carried into C under a transforma­ tion of the form (3.37). Hence the parametric equation of a line is i(x 0 0 [\0

0 0 0 0

0 0 0

y 1 1

x,yeD, 1/

(x,y)^(0,0)

(3.46)

132

Chapter 3. Geometry of Rectangular

where T € GLm+n(D).

Matrices

Write T in block form A T=( 1 \C

B

]

DJ>

where A = (a»j)l
=

B — (6ij)l<j
(Cti)l<*<m,l
D = (dtj)l
Then a point on the line (3.46) can be represented by /xau

+ ycu

xaln + ycln

xbn + ydn

C2n

d2l

C-mn

d,ml

xblm + ydim \ dim timm

(3.47)

where x, y G D and (x, y) ^ (0, 0). If the line (3.46) is contained in A4i, then the point (3.47) on it could be represented by / xau + yen

• • • xaln + ycin

xbn + ydn

\

\

1/

Therefore the parametric equation of a line in M.\ is f

/ a n • • • a i n bn

Cln dl

+»

x {

\

1/

\

1/

where ( a n , • • •, a l n , &n) ^ 0, ( c n , • • •, c a „, d u ) ^ 0, x,y € D, and (x,y) (0,0). Clearly, this line in Mi is mapped under the map (3.43) into {x(an,

• • •, aln, i n ) + y(cn,

• • •, c l n , d u ) | * , 2/ € A (x, y) ^ (0, 0)} ,

which is the parametric equation of a line in PGl(n,

D).

^

3.6. Application to Geometry

133

Now suppose that the line (3.46) is contained in M^. At first, we give an equivalent definition of the map (3.45). It is clear that the image of the point / l 0 ••• 0 0 0 ••• 0 0 0 ••• 0 \ 0 0 ••• 0 1 x2,n+i 0 0 ••• 0 1 xz>n+i 0 0 0 0 \0

0

-*•

*E*,n-f-»

1 •••

0

1/

in M2 under (3.45) is the point in PGr(m, D) whose first coordinate is zero and the last m coordinates form a nonzero solution vector of the system of linear homogeneous equations

(0 0 • • 0 1 1

X2,n+i

*1 X2

X3,n+i

X3

J-

^t,n+t

Xi

0

\

/

\

= 0.

Xi+1

I)

V

\Xm

)

Hence we deduce that for any point in M.2, whose homogeneous coordinate is of the form O^'71"1)

(x

F),

where x is an m-dimensional nonzero column vector and Y is an m x m matrix of rank m — 1, its image under (3.45) is the point in PGr(m, D), whose first component is zero and the last m coordinates form a nonzero solution vector of the system of linear homogeneous equations

Y

x2

\xmJ

= 0.

Chapter 3. Geometry of Rectangular

134

Matrices

Since the line (3.46) is contained in Mi, we have / Gl2 Cl2

\ Cm2

Gl3

' **

c

•* •

13

<*>ln \ Cl*

= 0.

cmn /

Cm3

Therefore the line (3.46) becomes < I i o n + j/cn

IV

0

c2i

0

Cral

0

••• 0 •••

xbu + ydu

• • ■ xhm + j/d I m \ ~j

d2\

0

•■■

d2m

(3-48)

/ J

where x, y £D, (x,y) ^ (0,0). Since T £ GLm+n(D), m. But now (cu (C D)

},

Jn

wehaverank(C D) =

0

••• 0

C21

0

•••

0

C?21

^22

d12

dim \ dim

Vc m l

0

•••

0

dmi

d,ml

(*mm /

thus rank A n - i > ra — 2, where ^21

^22

•••

dim

Dm-l

< dm\

dm2

We distinguish the following two cases. (a)

rank Dm-\

= m — 1. Since

/an

T =

0

•••

0

6n

#2i

Onl

«n2

Cli

0

0

C?n

C?i2

C21

0

0

C?21

^22

0

0

dml

d m2

\cml

22

•••

a2n

^

21

612

a

^

22

fcm\ &2m

Km dim dim

€ GLm+n(D),

* * * dmm )

3.6. Application to Geometry

135

at least one of D=(du

dl

>

•'"

dlm

)

(bn

and

bl

*

'•'

blm

)

is of rank ra. Let us consider the case when rank D = m. (The other case can be treated in a similar way.) Then m

(&llj 6l2, ' * • , bim) = $^r*(*fo>-<*t2> ' * * , dim),

where rt- 6 D (i = 1, 2, • • •, ra), and the points on the line (3.48) can be represented by ( zx C21

\Cml

0

0

0

0

C?21

<^22

0

0

%1

*m2

(xri + y)dn

(xri + y)d12

(xr a + y)di m ^

(3.49) m

r c

c

where zxy = x(au— J2 i n) + y iii

x

, y 6 D and (x, y) ^ 0. We distinguish

t=2

the cases xri + y ^ 0 and xri + y = 0. (a.l)

Let D

xri+y

x

^ 0. Then the point (3.49) can be represented by 1 / (*ri +c 2y)"" *** i

0

V

0

Cml

• • • 00 0

\

D I

= (Pij)i
0 0

\

1 1

(3.50)

u where tx = (xri + y) 1zxy

and

U = 5^p»j c ii (* = !> 2 , • • •, m). i=2

Given any element u in the division ring D, it is easy to prove that there are elements x, y G D with (x, y) ^ 0 and xri + y ^ 0 such that w =

Chapter 3. Geometry of Rectangular

136

Matrices

(xri + y) 1zxy. Thus the element u in (3.50) runs through the division ring D. Moreover, it is clear that for two nonzero 2-tuples (#, y) and (#i, y\) with (xu t/i) = l(x, y), where I G D*, we have (xrx + y)~1zxy = ( x i r i + y i ) " 1 ^ ^ , i.e., we get the same u. Under the map (3.44), the point (3.50) is mapped into the point /

1

\

puu + h P21U + h \PmlU + lm/

of PG r (m, D). (a.2) xri + y = 0. Then zxy ^ 0 and there is only one point on the line (3.49) satisfying this condition. Such a point can be represented by 1 0

0 • • 0 0 • • 0

0

0 • • 0

0

0

•••

o\

An-1

We have (Pll\

/0 V

0 ••• An-l

0\ )

P21

= 0.

\PmlJ Thus under (3.45) the point is mapped into the point / 0 \ P11 P21

\PmlJ of PGr(m,

D).

Assembling (a.l) and (a.2) we conclude that in Case (a) the line (3.48) is

3.6. Application to Geometry

137

mapped under (3.44) and (3.45) into / 0 \ Pu

1

(

)

P21

h u+ h

\Pml/

\L)

<

V

u,veD,

(u, v) ^ (0, 0)

►,

which is a line in P G r ( m , D). (b)

rank Dm-i

= m — 2. We can assume that

rank

/ d23

d24

'

^33

d34

•

\4

V «m3 3

«m4 dm4

•'

d2m \ 4m

' * * dmm dmm

= m -- 2 ,

''

(since the other cases can be treated in a similar way). Then

(

<*31

]

m

( d& )

=E \dmJ

»=3

/ S{

and

\dmi)

d22\ 42

( d$i )

m

=£ \dm2/

»=3

U.

\dmi/

Thus /xbn + yd fn 1 d2\ V

dmi

xb12 + yd12 d22 d>m2

• • • xbim + ydlm \ -" d2m '''

dmm

/

/

u -53U -

\ - S

m

U - *

\ £3t>

m

U /

0 0

7

UJ

where z =x(bn

- £ bnSi)u + y(dn - £ t=3

duSi)u

i=3 m

m

+x(bl2 - £ buti)v + y(d12 - £ t=3

duU)v.

t=3

For any (x, y) ^ 0, the following equation 771

771

X (&11-X) biisi)u+y(dn-Y^ «'=3

! >+y(di2-£] C?xi
b lti t

duSi)u+x(b12-^2 «=3

m

771

t=3

»'=3

Chapter 3. Geometry of Rectangular

138

Matrices

has a nonzero solution (u, v) ^ 0, and conversely. Therefore the line (3.48) is mapped under (3.44) and (3.45) into 0 0 1

0 1 0

U+

S3

^\-s

m

/

w, v G -D, (w, v) ^ 0

"*3

\-tm/

which is a line in PGT(m,

D).

D

From Proposition 3.39 we conclude that the maximal set of rank 1 M\ (or M2) has an n-dimensional left (or m-dimensional right, respectively) projective space structure over D. By Proposition 3.35, any maximal set of rank 1 is equivalent to M\ (or M.2) under the group PGLm+n(D). Therefore any maximal set of rank 1 equivalent to Mi (or M2) under PGLm+n has an n-dimensional left (or m-dimensional right, respectively) projective space structure over D. Parallelly, we can define the right Grassmann space of (n — 1)-dimensional flats in the (m+n — l)-dimensional right projective space PG fr (m-f n — 1, D), which is also called the projective space of (m + n) x n matrices over D and denoted by VM(m+n)Xn(D)We can define the adjacency, arithmetic distance, distance, and maximal set of rank 1 in a similar way and we have Propositions and Corollaries 3.28 - 3.39 for VM(m+n)Xn(D) also. Moreover, by the principle of duality we have a bijection VMmx(m+n)(D) W

VM(m+n)xn{D)

with the property

( r ) 1 = w. We recall that WL = {'v € PGr(m + n - 1, D)\w*v = 0 for all w G W} and (WL)L

= {ue

PG\m

+ n - 1, D)\u*w = 0 for all *w G

WL).

3.7. Application to Geometry (Continued)

139

Furthermore, we have P r o p o s i t i o n 3.40: Let W\ and W2 be two points of VMmX(m+n){D). Then / x L ad(Wi, W2) = ad(W 1 , W2 ). In particular, W\ and W2 are adjacent if and only if Wi" and W2 are adjacent. Proof: Suppose that ad(Wi, W2) = r (0 < r < m). Then Wx U W2 is an (m + r — l)-flat. By the dimension formula, i.e., Proposition 2.14, dim(Wi PI W2) = dim Wi + dimW2 - dim(Wi U W2) = m — r — 1. By Proposition 2.19 Wf U W2L = (Wi 0 W 2 ) x and dim(Wi fl W2)L = n + r - 1. Hence Wf U Wj 1 is an (n + r - l)-flat in PGr(m + n - 1, D). ad(W 1 1 , W2L) = r.

Therefore □

C o r o l l a r y 3 . 4 1 : Let .4 be a bijective map from VMmX(m+n)(D) to itself and assume that both A and A'1 preserve the adjacency. Then A induces a bijective map A* from VM(m+n)xn(D) to itself in the following way A*:WL^->A{W)L

ioiallWeVMmx(m+n)(D).

Moreover, both A* and A*-1 preserve the adjacency. Proof: Let Wu W2 G VMm><{m+n)(D),

then

ad(WV. W^) = ad(Wi, W2) = ad(^(Wk), .A(W2)) = a d ^ t t ^ , ^(W2)x) = a d ( . 4 W ) ,

A'(W£)). D

3.7

Application to Geometry (Continued)

The transformation of VMmx(m+n)(D) as follows.

of the form (3.37) can be generalized

Chapter 3. Geometry of Rectangular

140

Let a be any automorphism of D and T € PGLm+n(D). transformation on VMmx(m+n)(D): Wt

W°T

for all W €

Matrices

Define the following

VMmx(m+n){D).

It is clear that (3.51) is a bijective map of VAimx(m+n)(D) the arithmetic distance.

(3.51) and preserves

When 77i = n, we define a new type of transformations in the projective space of n x 2n matrices VMnx2n{D). Let N be a 2ra x 2n invertible matrix over D and r be an anti-automorphism of D. For any point W G VM.nx2n(D), define W^r = {ve D^\vN\WT) = 0}. (3.52) Since rankVK = n, we have also rank*(W T ) = n. By Theorem 1.14(ii), r a n k i V * ^ ) = n. Then by Theorem 1.29, d i m W ^ r = n. Therefore W^T is also a point of VA4nx2n{D). Clearly, we have

wjftTNt(ur) = o. Proposition 3.42: For any 2rc x 2n invertible matrix N over D and antiautomorphism r of Z), the map W

Wh N,T

(3.53)

is a bijective map from VMnx2n(D) to itself, and preserves the arithmetic distance between any pair of points in VMnx2n{D). Proof: Let W and V be two points of VMnx2n(D) T

V#tT. We have Wjj%TN\W )

T

and assume that W^

= 0 and V^fTJV*(V ) = 0. Then W^TN

CY

= 0.

From rank W^T = n, we deduce rank

f(v)1


=

3.7. Application to Geometry (Continued)

141

and then rrank a n k((£^ ) < < n».. Since both W and V are of rank n, there is an element P G GLn(D) that V = PW, i.e., W and V represent the same point of VM.nX2n(D)proves that the map (3.53) is injective. Let W be any element of VMnx2n{D).

Define

T 1 T r 1 G JD^\v\N ) W | |u'(AT VT = " = {v {t, G W* ~ )\W"T"~1)'(W )

■ ' )

Then W* is a point of VMnx2n(D)

such This

0}. == 0}. =

and

T \NT~~ )\W "1)'(W 0. W* (N ~ ) "") = ==Q. w* t

T 1

r 1

Applying r and the transpose to the above equation, we obtain r T) := 0. WN \W* WNi(W* ) = 0.

Therefore under the map (3.53) we have

w*

1 — ►

w.

This proves that the map (3.53) is surjective. Now let W and V be two points of VMnX2n(D) and assume that ad(W, V) = r, then by Proposition 3.30, there is an element T G PGL2n(D) such that W = (0 lW)T and V = (Er /( n ))T, where

Er: Let

/JM " V0

0>

)

n=

tT T W^ (X* Y*) == WJv, ) ) TN (T rN\T

and T

(Z* Vir\T N\T),T), (Z* IT) U*) ==-■v^ rN where X*, Y*, Z*, and U* are n X n matrices, then (n)

{X* yr*)'(o (X* ) ' ( 0 7J(n) )) = == 00

142

Chapter 3. Geometry of Rectangular

Matrices

and n) (Z* u*y{E (Z* U*Y{Er r jJ( n( >) ) === 0.

Consequently, F * = 0 and Z*Er + U* = 0. Therefore rank ^ ^* »

ank f ^ U* )= = rrank JJ*) ( Z*

_Z*E -Z*E

n + rank rank(Z*£' < rc + + r. r. )) == n + ( ^ * ^ r))
Hence rank (( ^ ^ rank ( y^T ) == rank - ■ * ( $ ; ) ■

z>

rank f y J • ^ J < nn ++ rr =-=rank (v) •

By symmetry, we have the proposition.

□

Proposition 3.43: The product of two transformations of VMnX2n{D) the form (3.53) is a transformation of the form (3.51).

of

Proof: Let Ni and AT2 be two 2n x 2ra invertible matrices over D and T\ and T2 be two anti-automorphisms of D. Then W Whl—» - - Wife ,„

for VM for all all W € P;W (i)) Mm(D) n><2n

(3.54) (3.54)

and W

Wy-^ -

W

kr, forfor all WiU

W € VMVM W n^n{D) n*2n{D)

(3.55)

are two transformations of the form (3.53). We have t Wj? W ^ltT1 ^1i (W*) '(W^) = =0 1 (N

(3.56)

< W£ \Ur>) ^4,r (^ T 2 ) = 0. 0. 2tnN22^

(3.57)

and

Applying r 2 and the transpose to (3.56), we obtain |^TlT2

rtw,,n D == 0.

*(JVJ>

Thus 1 n W™ ) liTl )T2) = W™ \Np)N \N?)N?N *((W^ = 0. a- 2N2\{W^)

From (3.57) and (3.58) we deduce — \ynT2 (Wb, = W™\N?)N12-\ 2 TiJN2,T2 W i , n)kr

WW -

(3.58)

3.7. Application to Geometry

(Continued)

143

Thus the product of the transformations (3.54) and (3.55) is W

>_

Wnr2

' ( A ^ J V - 1 for all W G VMnx2n(D).

(3.59)

It is clear that the product T\T2 is an automorphism of D and t(Nl2)N2~1 G GL2n(D). Therefore (3.59) is a transformation of the form (3.51). □ Corollary 3.44: Let m = n and assume that the division ring D has antiautomorphisms. Then the group of transformations of Vnx2n(D) generated by all transformations of the form (3.51) and of the form (3.53) can be gen­ erated by all transformations of the form (3.51) and a single transformation of the form (3.53). □ We usually take the transformation W »—►

w

£,r

for

all W G VMnx2n(D),

(3.60)

where r is an anti-automorphism of D and

<3-61)

*=(_£»> o )'

as one of the generators of the group of transformations of Vnx2n(D) gener­ ated by all transformations of the form (3.51) and of the form (3.53), or we take the transformation W H-> Wf{2n)yT for all W G

VMnx2n(D)

as one of the generators of the group. The goal of the present section is to prove the fundamental theorem of the projective geometry of m x (ra + n) matrices over any division ring, which reads as follows. Theorem 3.45: Let D be a division ring and m and n be integers > 2. Let Abe a, bijective map from VA4mx(m+n)(D) to itself and assume that both A and A'1 preserve the adjacency of pairs of points in VA4mx(m+n)(D). Then, when m ^ n, A is of the form (3.51) W

, — WT

for all W €

VMmx(m+n)(D),

144

Chapter 3. Geometry of Rectangular

Matrices

where a is an automorphism of D and T G GLm+n(D). When m = n, A is either of the form (3.51) or the composite of a transformation of the form (3.51) and the transformation (3.60) W forall allWW €G7>.M„x2n(£>), VMnx2n{D), ^ .— ^ ► ^W £ T for where r is an anti-automorphism of D, K is (3.61), and W^T is defined by (3.52). Conversely, any map of the form (3.51) (or of the form (3.60), when m = n) from P M m x ( m + n ) ( ^ ) to itself is bijective and both the map and its inverse preserve the adjacency. To prove this theorem we need some preparation. Assume that both m and n are integers > 2. For any W G write W = (W (wi1 W w22 W =

• ' -* •"

VA4mX(m+n)(D),

Wm+n),

where W{ is the i-th column of W (1 < i < m + n). Let 1 < i\ < i2 < • • • < im < m + n and define Mili2 Mi li2...im m

= ={W {W G e PA^ VMmxim+n^D^iwi, -2 •• ' •• • wim) wim) ls isinvertible}. invertible}. mX (m+n)(I>)|(ix; il wti2

Multiplying each point W G Mi^...^ by (wi:L wi2 ••• w t m ) - 1 from the left, we see that W can be represented by an m x n matrix over D (whh (w

1 w wi2i2 ■••• ■■ ww imim ) )~ {w * Kh i

whj2 ■••• jn), ■• w Wjn),

w

(3.62)

where 1 < jx < j 2 < • • • < j n < m + n and iu i 2 , • • •, i m , j i , j 2 , • • •, jn is a permutation of 1, 2, • • •, m + n. We call (3.62) the non-homogeneous coordinate of the point W G «A"ftii2-tm- Clearly, we have L e m m a 3.46: Let m and n be integers > 2. Then the map mXn mXn W ^

i—► ^ K(to,-,

(D)

wi2*2 '•• ■ ■■ ■W ™ «iJ0

W

■■ ™jn) t»i„) V H^Jl i l Wh h ■'••

is bijective and both the map and its inverse preserve the adjacency of any pair of points. □ L e m m a 3.47: Let m and n be integers > 2. Then

VMmx{m+n) (D) = ?A< mx(m+»)(5)

(J |J 1<«1<*2 <-
Mili2 M...,im lt-2... m. D

□

3.7. Application to Geometry

(Continued)

145

Moreover, we have L e m m a 3.48: Let m and n be integers > 2, and A be a bijective map from M>mxn(D) to itself. Assume that both A and A'1 preserve the adjacency of pairs of points of Aimxn(D). Let (i 0 , jo) be a fixed pair of indices, where 1 < zo < m and 1 < jo < TI. Assume that A leaves every point whose (io, jo)-entvy is nonzero fixed, then A leaves every point of MmXn{D) fixed. Proof: Consider only the case m < n, for the other case m > n can be treated in a similar way. Without loss of generality we may assume that i0 = m and j 0 = 1. We have to prove that A leaves also every point whose (m, l)-entry is zero fixed. Let /

zn

x

X\2

ln

\

W = Em-1,1

Xm-1,2

0

Xm2

V

%m—l,n

By hypothesis, we can assume that /

x\x

12

*i»

\

A{W) = m—l,n

V o

X

L

mn

m2

I

If x\\ 7^ 0, then for any A2, • • •, Am G -D, / x n

X12

•••

xi

m

£i,m+i

#ln\

A2

G -M mX n(/?). V 1

/

Am

By hypothesis, ^(VKi) = W\. Clearly, ad(VK, W\) < n. Therefore we have also ad(^4(H^), W\) < n. That is, the matrix

/x^-xu L

21

x\2-x12 ^22 - A 2

<-771-1,1

X m-1,2

-1

X m2

Xlr<

^+1 L

^771

^ra-l.m X

mm ~ A„

L

^l,m+l

2,Tn+l

m-l„m+l ^TT^TTI+I

X

x

ln

ln

b

2n

z;77i—l,n

\

146

Chapter 3. Geometry of Rectangular

Matrices

has rank < n. But A2, • • •, Am are arbitrary elements of D, so this is possible only when x\x = x n . If rrjj ^ 0, then applying the above argument to *4 _1 , we obtain also x\x = Xu. Therefore we always have x\x — X\\. Similarly, we can prove that x*j = xtJ- for all z, j . □ Now let us come to the proof of Theorem 3.45. Proof of Theorem 3.45: Consider first the case m < n. Let A be a bijective map from VM.mX(rn+n)(D) to itself and assume that both A and A'1 preserve the adjacency of pairs of points in VAAmX(m+n)(D). By Corollary 3.33, A also preserves the arithmetic distance between any pair of points in VMmx(m+n)(D). We proceed in steps. (i) We distinguish the following two cases. (a) m — n. By Proposition 3.28, after subjecting A to a transformation of the form (3.37) we can assume that .4(0 / W ) = (0 J). Clearly, M\%.). Thus A{Ml2...n)

= Mi2.-*.

Let Z be the non-homogeneous coordinate of a point in AAu-n- By Lemma 3.46 and Theorem 3.4, we have either A(Z) = PZ"Q + R

for all Z € -M12...n,

where a is an automorphism of D, P , Q £ GLn(D), A{Z) = P \ZT)Q

+ R

(3.63)

and R G Mn(D),

for all Z G M i 2 . . , ,

or (3.64)

where r is an anti-automorphism of D. In the first case, expressing (3.63) in homogeneous coordinates, we have A(X

Y) = (X Y)' ( P _ 1

P

li? g

)

for all (X Y) € -M12...„.

Thus after subjecting A to the transformation of the form (3.51)

(X Y)~\(X F)(

P_1

7 P R -IT

2 )

for all (X Y) €

PMnx2n(D),

3.7. Application to Geometry (Continued)

147

we can assume that

A{X ----{X Y) for for all A{X Y) = (X Y) all (X {X Y) € GM M1212... ..«. n. In the second case, expressing (3.64) in homogeneous coordinates, we have

A(X Y)kTtT A{X Y) = (X Y)k,

for for all (X {X Y) Y) € M M12 ..«, n, 12...

where l

» - ( - &

".')•

Thus after subjecting A to the transformation of the form (3.53)

(X ) _- * ( ^ (x yY)^(X

y)V-'),r-i> )t(W- ),T-l' y

1

we can also assume that A(X Y) ( * Y) Y) ^(x y) === (X

for all for all (X K Y)) € e M\2-ixM12...n.

(b) m < n. A l l is a maximal set of rank 1, so A(A4i) is also a maximal set of rank 1. By Proposition 3.35, A(M.\) can be carried under a transformation of the form (3.37) to either Mi or M.2> By Proposition 3.39, M\ has an n-dimensional projective space structure, so does A(M\). But M2 has an m-dimensional projective space structure. By the fundamental theorem of projective geometry A{M\) cannot be carried to M2 under a transformation of the form (3.37). Then after subjecting A to a transformation of the form (3.37), we can assume that /('xX\l n

X12 x\2

••*• •

N \

£i,n+i a;i,n+i

1 1

A

1) 1/

\ /( *x\i ix

X{ 1 22

x

'•

•• '

*

\\

x x l,n+l

l,n+l

1 1

}

1

I V

lj 1/

Chapter 3. Geometry of Rectangular

148

Matrices

where ( x ^ a^2 • • • # l i n + i ) = ( £ i i £12 • • • £i, n +i)" Q, & is an auto­ morphism of D, and Q € GLn+i(D). Thus after subjecting A to a bijective map of the form (3.51) W \Q~'

W\

/(m

-i))f

f^ all W €

VMm><{m+n)(D),

we can assume that A(W) = W

for all W G

ML

In particular, the set of points /0

•••

0

Xi,m+l

•*•

£l,n+l

\

1 1

(3.65)

1/J

I \

where £i >m +i, •••, Xijn+X G D and (#i, m +i, •••, a?i,n+i) 7^ 0, is left fixed elementwise by A. Clearly Mu-m consists of all the points which are of arithmetic distance m with every point of (3.65). Thus A(M12...m)

= M12..*.

As in Case (a) we can assume also that A(X (ii)

Y) = (X Y)

for all (X Y) G M12...n.

Define S = {(w1 0 ••• 0 wm+2

•••

wTn+n)\w1^0}nVMmx{m+n).

We assert that A(S) = S. Let

W =

( Xu

0

•••

0

Xhm+2

''*

Zl,m+n \

X2X

0

•••

0

X2,m+2

•' '

X2,m+n

U

&m,m+2

" * '

\Xml

0

%m,m+n)

eS

3.7. Application to Geometry

(Continued)

149

and assume that ,4(1^) $ S. It is easy to see that there exists a point in -Mi2-.m, whose arithmetic distance with W is < m. Assume that / xn A(W)

x12

=

\ Xml

x

^22

^21

X

m2

2,m+n

X

' ' '

m,m+n

I

If ^11

L

12

C

C

22

21

\*ml

•

*i

\

m

b

2m

o,

X*nn I

<

mm /

then every point in M\2>.m is of arithmetic distance m with »4(VT). Since A leaves M.\2-..m fixed elementwise, we get a contradiction. Therefore 1x p

n

^12

21

C

\Xml

\

22

^2771

7^0.

^7712

Since ^ ( W ) ^ 5 , we have / x\i

x

\z

X

^1,771+1

X

22

C

2Z

\ < 2

< 3

•••

2,77l+1

^0.

J

<,m+l.

We can choose a matrix representation of *4(VK) such that \XW>

x

' ' ' •> Xlm)

\2)

T 0

and ( x 1 2 , £ 1 3 , • • • , ^ i > m + i ) 7= 0.

There exists a point I

W' =

^11

X

12

*' '

W21

U22

' "

\ Wml

U>m2

^1,771+71

\

^m+n

^m,m-\-n

/

150

Chapter 3. Geometry of Rectangular

Matrices

such that both / xu

x12

---

\ Um\

Um2

'''

xlm \

Umm I

and / x12

x13

-••

U22

^23

* * • V>2,m+1

\ Um2

W m3

•••

are invertible. Then W G M^-m

xlm+1

Um,m+l /

and A{W)

r a n k

( ^ ' )

\

= W.

Clearly,

= 2 m

and raak(^))<2m. This is a contradiction. Therefore A(S) C S. Similarly, we can prove that A(S) D S. Hence A(S) = S. (iii) -M23-m+i consists of all the points which are of arithmetic distance n, with every point of S. Hence A(M23...m+l) = A423...m+1 and by Lemma 3.46 A induces a bijective map from A<23- -m+i to itself such that both A and A'1 preserve the arithmetic distance between any two points of A^23.-m+i- The set of points [ / ^11 #21

1 1

£l,m+2

•••

Xi,m+n \ I

£2,m+2

•••

Z2,m+n

:

I x

V \ ml

I ' -»-

x

m,m+2

'''

x

m,m+n / )

where Xij e D (i = 1, 2, • • •, m; j = 1, m + 2, • • •, m + n) and x m i ^ 0, lies in A
3.7. Application to Geometry (Continued)

151

By Lemma 3.47, VMmx(m+n)(D) is the union of all the A^tlt2...,m, where i\, ^2, • * *, im run through all the m-subsets of {1, 2, • • •, m + n} such that 1 < ii < i 2 < • • • < zm < m + n. Therefore A(X) = X

ioval\X€VMmx{m+n)(D).

Now consider the case m> n. Let A be a bijective map from VMmX(m+n) (D) to itself and assume that both A and A'1 preserve the adjacency of pairs of points in VMmX(m+n)(D). By Corollary 3.41, A induces a bijective map A* from VM(m+n)xn{D) to itself, where A* :

wL - - » A{W)L

for all W €

VM(m+n)xn{D),

and both A* and A*~x preserve the adjacency. We have n < m. parallel to the proceeding proof we can show that A* is of the form A*{WL)

=

T(WLY,

where a is an automorphism of D and T 6 PGLm+n(D). A(W) = (A^W1))1-

= (T(WLy)L

Then

= (TiW*)^

Then =

W°T-\

□ Parallel to Theorem 3.45 we have also proved T h e o r e m 3.49: Let D be a division ring and m and n be integers > 2. Let A be a bijective map from VM(m+n)Xn(D) to itself and assume that both A l and A~ preserve the adjacency of pairs of points in VM(m+n)xn(D). Then, when m ^ n, A is of the form W

_

T W " for all W € P M ( m + „ ) x » ( 0 ) ,

(3.66)

where a is an automorphism of D and T G GLm+n(D). When m = n, ^4 is either of the form (3.66) or the composite of a transformation of the form (3.66) and the transformation W »—> Wk%T for all W G VM2nxn(D),

(3.67)

Chapter 3. Geometry of Rectangular

152

Matrices

where r is an anti-automorphism of Z), K is (3.61), and Wk%r = {'* € tD^n)\\WT)Kiv

= 0}.

Conversely, any map of the form (3.66) (or of the form (3.67), when m = n) from VM(rn+n)xn{D) to itself is bijective, and both the map and its inverse preserve the adjacency. O Let us generalize the map (3.53) as follows. Let D and D' be division rings, r be an anti-isomorphism from D to D', and N be a 2n x In invertible matrix over D'. For any point W G VMnx2n, define Wj(fiT = {ve D'^\vN\Wr)

= 0}.

Similar to Proposition 3.42 we can prove that the map W >-> H# i T is a bijective map from VMnX2n{D) to VMnx2n(Df) and preserves the arith­ metic distance between any pair of points. Without essential difficulty, The­ orem 3.45 can be extended as follows. Theorem 3.50: Let D and D' be division rings, m, n, m' and n' be integers > 2, and VMmx{m+n)(D) and / PA^ m / x (m / +n , )(^ )/ ) be the projective spaces of m x (m + n) matrices over D and of mf x (772' + n') matrices over Z);, respectively. Let A : / PA4 mX ( m + n )(D) 1—> VMm'x{m'+n')(D') be a bijective map and assume that both A and A'1 preserve the adjacency of pair of points. Then m = m' and n = n'. When m / n, Z) is isomorphic to D' and A is of the form W^WT

iox*\\WeVMmx(m+n){D),

(3.68)

where a is an isomorphism from D to D', and T G GLm+n(D'). When m = n, A is either of the form (3.68) or the composite of a transformation of VMnX2n{D') of the form (3.51) and the transformation from VMnx2n(D)

to

VMnx2n(D') W >—► W £ T

^ all W G VMnx2n(D),

(3.69)

3.8. Application to Graph Theory

153

where r is an anti-isomorphism from D to J9', K is (3.61), and

W^T = {ve D'^\vK\WT)

= 0}.

Conversely, any map of the form (3.68) (or of the form (3.69), when m — n and D is anti-isomorphic to D') is bijective and both the map and its inverse preserve the adjacency. □

3.8

Application to Graph Theory

Let D be a division ring, and m and n be integers > 2. Now we call the points of MmXn{D) vertices and define two vertices Z\ and Z 2 to be adjacent if rank(Zi — Z 2. Then the graph T(MmXn(D)) is connected, distance-transitive, and of diameter min{m, n } . Q Theorem 3.21 can be interpreted as follows. Theorem 3.52: Let D and D' be division rings, and m, n, m' and n' be integers > 2. If there is a graph isomorphism from the graph T(M.mXn(D)) to r(A^ m / X n'(^ / ))5 t n e n either m = ra', n — n', and D is isomorphic to £)', or m = n', n = m', and D is anti-isomorphic to D'. In the first case, the graph isomorphism is of the form (3.29) and in the second case, the graph isomorphism is of the form (3.30). Conversely, if a is an isomorphism from D to D\ and m = m', n = n', then the map (3.29) is a graph iso­ morphism from the graph r ( A ^ m X n ( ^ ) ) to T(Mm'xn'(D')), and if r is an

Chapter 3. Geometry of Rectangular

154

Matrices

anti-isomorphism from D to D\ and m = ra', n = n', then the map (3.30) is a graph isomorphism from r(wM m X n(£)) to T(MmiXni(D')). □ Corollary 3.53: Let D be a division ring and ra and n be integers > 2. If ra ^ ra, then the group of graph automorphisms of the graph T(Mmxn(D)) consists of the following graph automorphisms X i—► PXaQ

+ R

for all X G T(Mm*n(D)),

(3.70)

where a is an automorphism of D, P G GLm(D), Q G GLn(D), and i? G A4 m xn(£))- If ra = ra, then the group of graph automorphisms of the graph T(Mn{D)) consists of the graph automorphisms of the form (3.70) and the graph automorphisms of the following form X h-+ P\XT)Q

+ R

for all X G T(Mn(D)),

where r is an anti-automorphism of D.

(3.71) □

Similarly, call the points of VMmx(m+n){D) vertices and define two vertices W\ and W2 to be adjacent, if V^i U W2 is an ra-flat. Then we obtain the graph of the left Grassmann space VA4mX(m+n)(D)- Denote this graph by {m+n){D)). From Proposition 3.31 and 3.32, we deduce immedi­ ately Proposition 3.54: Let D be a division ring, and ra and n be integers > 2. Then the graph T(VM.mX(m+n)(D)) is connected, distance-transitive, and of diameter min{ra, n } . □ Theorem 3.50 can be interpreted as follows. Theorem 3.55: Let D and D' be division rings, and ra, n, m' and n' be integers > 2. If there is a graph isomorphism from T(VMmX(m+n){D)) to T(VMm'X(m'+n'){D')), then ra = ra; and n = n'. When ra ^ n, D is isomorphic to Z?' and the graph isomorphism is of the form (3.68). When ra = ra, the graph isomorphism is either of the form (3.68) or is the composite of an automorphism of VMnX2n{Df) of the form (3.51) and the isomorphism (3.69). □ Corollary 3.56: Let D be a division ring, and ra and ra be integers > 2. If ra 7^ ra, then the group of graph automorphisms of the graph of the

3.9.

Comments

155

left Grassmann space T(VMmX(m+n)(D)) automorphisms W ,—> PWT Wt-> PWT

consists of the following graph

for all all W W G G T(VM (D)), (D)), for T(VM mx{m+n) mx{m+n)

(3.72) (3.72)

where a is an automorphism of D, P G GLm(D), and T G GLm+n(D). If m = n, then the group of graph automorphisms of the graph T(VMnx2n{D)) is generated by automorphisms of the form (3.72) and the automorphisms Wk,r,T, - Wk,

W W H•—

where r is an anti-automorphism of D, K is (3.61), and W^T (3.52).

3.9

(3.73) (3-73) is defined by □

Comments

The concept of the adjacency (or coherence) of a pair of m x n matrices in Section 3.1 is due to L. K. Hua, cf. Hua 1951. Theorem 3.4 for any division ring D ^ F 2 is due to L. K. Hua, cf. Hua 1951 and for the case D = F 2 is supplemented by Z. Wan and Y. Wang, cf. Wan and Wang 1962. All the results of Section 3.2 are due to L. K. Hua, cf. Hua 1951. In particular, the maximal set of rank 1, which was introduced and called the maximal clique in graph theory in the seventies, was introduced by Hua in 1951. All the results of Section 3.3 are due to Z. Wan and Y. Wang, cf. Wan and Wang 1962. The proof of Theorem 3.4 when D ^ F 2 in Section 3.4 is adopted from Hua 1951 and the proof for the case D = F 2 is adopted from Wan and Wang 1962. The results of Section 3.5 are immediate generalizations of the results of L. K. Hua on the automorphisms, semi-automorphisms, and Jordan automor­ phisms of Ain(D), cf. Hua 1951. Hua also determined the Lie automor­ phisms of Mn(D) when n > 2 and D is of characteristic different 2 and 3, cf. also Hua 1951.

156

Chapter 3. Geometry of Rectangular

Matrices

The fundamental theorem of the Grassmann space over a division ring (i.e., Theorem 3.45) is due to L. K. Hua, cf. Hua 1951. A sketch of the proof of the theorem was given in Hua 1951. Later a complete proof was provided by D. Pei, cf. Pei 1964. The contents of Section 3.6 and 3.7 are adopted from Pei 1964 but with some modifications. When D = F is a field, Theorem 3.45 was proved by W. L. Chow, cf. Chow 1949. From Chow's Theorem S.-T. Deng and Q. Li deduced Theorem 3.4 for the case D = F being a field, cf. Deng and Li 1965. The results of Section 3.8 are merely translations of the corresponding results of the previous sections into the graph theory language.

Chapter 4 Geometry of Alternate Matrices 4.1

The Space of Alternate Matrices

Throughout Sections 4.1 - 4.3 let F be a field of any characteristic and n be an integer > 4. We are now going to study the set of all n x n alternate matrices over F , which is called the space of n x n alternate matrices over F and denoted by fCn(F). The n x n alternate matrices over F are called the points of the space. With the space Kn(F) we associate naturally a group of motions which consists of transformations of the form X ^

tpXP + Ko for all X e lCn(F),

(4.1)

where P £ GLn(F) and K0 G Kn(F). Clearly (4.1) is a bijection. Denote this group by GKn(F). Parallel to Propositions 3.1 - 3.3 and Definition 3.1 of Chapter 3 we have Proposition 4.1:

GKn(F)

acts transitively on ICn(F).

□

Definition 4.1: Let X\ and X2 be two points in JCn(F). The arithmetic distance between X\ and X2, denoted by ad(Xi, X2), is defined to be the value of (rank(X! - X2))/2. If ad(Xu X2) = 1, i.e., r a n k ^ - X2) = 2, then Xi and X2 are said to be adjacent □ Proposition 4.2:

Let Xu X2, X3 e Kn{F). 157

Then

Chapter 4. Geometry of Alternate

158

Matrices

1° ad(Xi, X2) > 0; ad(Xi, X 2 )=0 if and only if Xt = X2. 2° ad(Xu X2) = ad(X2,

Xx).

3° a d ( X i , X 2 ) + ad(X 2 , X3) >ad(XuX3).

□

P r o p o s i t i o n 4 . 3 : The elements of the group GKn(F) leave the arithmetic distance between any two points of K,n(F) invariant. Moreover, for any r with 1 < r < [|], the set of pairs of n x n alternate matrices over F of arithmetic distance r forms an orbit under GKn(F). O We remark that from Proposition 1.34 we deduce Proposition 4.3 imme­ diately. Therefore the arithmetic distance is a geometric invariant under the group GKn(F), so is, in particular, the adjacency. Our purpose is to characterize the transformations of the form (4.1) by as few invariants as possible. We will see that the adjacency alone is almost sufficient to char­ acterize the transformations of the form (4.1) to within automorphisms of F. More precisely, we have the fundamental theorem of the geometry of alternate matrices over any field, which reads as follows. T h e o r e m 4.4: Let F b e a field of any characteristic, n be an integer > 4, and A be a bijective map from Kn{F) to itself. Assume that both A and A'1 preserve the adjacency, i.e., for any Xi, X2 G /C n (F), rank(Xi — X2) = 2 if and only if rank(,4(Xi) — A(X2)) — 2. Then when n > 4, A is of the form A(X) = plPX°P

+ K0

for all X G /C n (F),

(4.2)

where p G JP*, P G GLn(F), cr is an automorphism of F , and K0 G lCn(F). When n = 4, A is of the form A(X) = ptpWYP

+ K0

for all X G K4(F),

(4.3)

where X \—► X* is either the identity map or the map /

0

-x12 —«13

\-x14

X12

X13

X14\

0

x23

x24

-#23

0

X34

-x24

-x34

0 /

/

^ I

^

0

-x -X13

\-x23

u

X12

X13

X23\

0

x14

x24

-X14

0

#34

-x24

-x34

,

v

'

0 /

Conversely, any map of the form (4.2) or (4.3) from ICn(F) to itself is bijective and both the map and its inverse preserve the adjacency in K,n(F).

4.2. Maximal Sets

159

The proof of Theorem 4.4 will be given in Section 4.3. Parallel to Definition 3.2, Proposition 3.5, and Corollary 3.6 we have Definition 4.2: Let X, X' G Kn{F). When X ^ X', the distance between X and A ' , denoted by d(A, X'), is defined to be the least positive integer r for which there is a sequence of r + 1 points A 0 , X^ • • •, Xr with X0 = X and Xr = X' such that A, and X t +i are adjacent for i = 0, 1, • • •, r — 1. When X = A 7 , W e define d(A, X) = 0. □ For any two points A, A 7 G ICn(F)

Proposition 4.5:

ad(A, A 7 ) = d(A, A 7 ).

□ Corollary 4.6: Let A be a bijective map from fCn(F) to itself and as­ sume that both A and A'1 preserve the adjacency, then A also preserves the arithmetic distance, i.e., for any Ai, X2 G /C n (F), rank(Ai — A 2 ) = rank(^(A!)-^(A2)). □

4.2

Maximal Sets

Definition 4.3: Let M be a nonempty subset of Kn{F). Ai is called a maximal set if any two points of M. are adjacent and there is no other point of Kn(F) outside M, which is adjacent to every element of M. □ Proposition 4.7: Any maximal set in Kn{F) is carried into a maximal set under any transformation of the form (4.1). Q Proposition 4.8:

Both

r / Xi2 o

X\2

1 ~ I V

and

[(

1 j

***

\1

Xin

Ul2, •• • , «ln € F j

0 /

-Xln

0 -#12 -^13

^

1

(4.5)

J •v

x12

Zi3

0

^23

-^23

0

\ ^12, ^13, X23t

0/

F\

(4.6)

Chapter 4. Geometry of Alternate

160

Matrices

are maximal sets in Kn{F). Moreover, every maximal set can be carried under a transformation of the form (4.1) to (4.5) or (4.6). Proof: We prove only the second statement. Let Ai be a maximal set in Kn{F). By Proposition 4.3 we can assume that M. contains 0 and E\2~Ei\. 1 Let a\2

a13

•■

<*ln

0

«23

*•

0-2n

A=

\

o • *

0 / \ be another point of A4. We distinguish the following two cases. (a) a12 =^ 0 and 1. Since A and 0 are adjacent, rank A = 2. From ai 2 ^ 0 we deduce that the last n — 2 rows of A are linear combinations of the first two rows. Let a (—Git', GG t -tl-. lt.)t )0>0>a t\t'+l> ) ( — ^lt', —a — «2t' t\t'+l»* *' *' 5* 5a t nain) 2 t -,? •* •*• *, ,——

rz, n, = , • • •, «in) + A ( - « i 2 , 0, a 2323, , • •• •5• •,« 2an 2n i2, a ) j )j3 < = aatt(0, (0, aai2, a13 3 3, a2t- = —ojtai2 for i > 3, and atjb = otidik + Pia2k for k > i > 3. Thus a for aa>ik a 12a>i2( (aitaa2iiA; 2k•- —aikd>2i) a>ika>2i) tA; = =

k > i A>; >3. i > 3. for

(4.7) (4.7)

Similarly, since A and i?i 2 — E2\ are adjacent and a12 ^ 1 we have 1

Otik = (ai«a2jt -— auta 2t ) - «iik«2«) Otik = (ai2 ( « i 2 "■ - l)~ 1 ) l(a>iia>2k

for for

kk >> i i >> 3.3.

(4.8) (4.8)

Comparing (4.7) and (4.8), we obtain a,fc = 0 for all z, A; > 3. (b) aw = 0 or 1. We consider only the case a i 2 = 0, the case a i 2 = 1 can be treated in a similar way. Now (4.8) becomes atfc = —(ana2k — «iA?«2t) for k > i > 3. If an = a2t- = 0, then at£ = 0. Assume that an ^ 0. Since A and 0 are adjacent, rank A = 2 and the determinant of every 3 x 3 submatrix of A is 0. In particular 0 Olik ait

o —ait

<*2t

«2A:

0

AtA:

= 0.

1 The entries below the main diagonal in an alternate matrix are clear if the entries above the main diagonal are known, so sometimes they are not written out explicitly and expressed by * for abbreviation.

4.2. Maximal Sets

161

Consequently, aua2k — aikd2i = 0 and, hence, a^k = 0. For a2% ^ 0, the same argument shows also a,* = 0. Therefore in both cases A is of form /

0

#12

— Oi2

0

ai3 CL23

0>2n

0 *

o )

V Since A and 0 are adjacent,

rankf013 ■" M V «23

*' '

< 1.

0>2n J

Obviously the set 0 -x12

x12 0

(4.9)

*i2 € F 0,

is not a maximal set. Therefore there is an A G M. such that

rankf"13 0>2n n/

\«23

By Proposition 1.17 there exists an element P G GL2(F) Q e GLn_2(F) such that V a23

1 0 ,0 V 0

• • • a2n /

)

and an element

■

Of course we can assume that d e t P = 1. Then the transformation

x^'(P

Q)X(P

Q)

fora11

xeKn{F),

which is of the form (4.1), leaves both 0 and En — E21 fixed and carries A to / 0 a 12 1 0 ••• 0 \ -a12 0 0 0 ••• 0 0 (4.10) 0 \

0/

Chapter 4. Geometry of Alternate

162

Matrices

Therefore we can assume that M contains 0, E\2 — E2i, and (4.10). Let

/o

&12

&13

0

&23

B =

•

• • -

0 *

I

bln \ »2n

■

h-ln

0

)

be a fourth point in M- From a d ( 5 , 0) —aA(B, E\2 — E2\) = 1 we deduce as above that B is of the form (

0

&12

b\z

— b\%

0

&23

bln\ bin

0

B =

o / where rank ( &13 ' * ' hn \ < V ^23 • * * b2nJ ~ Since the subset (4.9) with the point (4.10) added is not a maximal set, there must be a point B such that

rankfj13 ' " [ln) = 1. Denote (4.10) by A. From ad(A, B) = 1 we deduce 624 = Hence /

0 -&12

&14 • • • bin \ 0 ••• 0

&12 &13 0 023

0

B =

0

0 /

\

where 623 = 0 or 614 = • • • = bln = 0. Clearly, the subset ( I

l\

0

X12

X13

-a; 1 2 -x13

0 0

0 0

\ a>i2, a;13 G F


= b2n = 0.

4.2.

Maximal

Sets

163

is not a m a x i m a l set, t h e r e m u s t b e a B with 623 7^ 0 or one with 611- 7^ 0 (4 < i < n). T h e points adjacent t o all t h e four points 0, E\2 — E2\, /

0 -b\2

b12

(4.10), a n d

bln\ with (614, • • •, bln) 7^ 0

0

V-6 In

/

m u s t b e contained in (4.5), and t h e points adjacent t o all t h e four points 0, £ 1 2 - £ 2 1 , (4.10), and /

0 —612 — &13

&i2 613 0 623 — &23 0

\ with 623 7^ 0

V

0/ D

m u s t b e contained in (4.6). Corollary 4.9:

[

^

x12

0 —x12 ':

tp I

A m a x i m a l set in JCn(F) is either of t h e form

• • • xln ^ P + K0 ^12, -•-, xine

0

\ -Xin

F

(4.11)

/

or of t h e form f

/ tp

0 —X12 —X13

£12 0 —X23

£13 £23 0

\ P + K0 X12, a;13, i 2 3 € F

(4.12)

0/

V where P G GLn(F)

and K0 G

Kn{F).

Let

{

/

0 -in

\

X12 0 *

a(a;i3, • • •, xln)' /?(xi 3 , • • •, Sin) X\2, • • • , 2ln G F > , 0

where ( a , / ? ) G F<2> and ( a , /?) ^ 0, a n d

i

f

V

0 -X12

X12 0

a;i 3 (ai, • " , 0 ^ - 2 ) ' 223(01, • • ■, an-2)

*

0

£12, Zi3, x 2 3 G F > ,

Chapter 4. Geometry of Alternate

164

Matrices

where (a 1? • • •, a n _ 2 ) G F^ n " 2 ) and (au • •, a n _ 2 ) 7^ 0. Then we have Corollary 4.10: The subsets M^p), where (a, £) G F& and (a, /?) ^ 0, and jV( ai ,..., an _ 2 ), where (au • • •, a n _ 2 ) G i^71"2) and (au • • •, a n _ 2 ) 7^ 0, are all the maximal sets containing both 0 and E12 — E21. Moreover, M[a,p) = M(a\p') if and only if there is a A G F* such that (a', /?') = A(a, /?), and A/(ai)...jan_2) = J\/( 0 ' ) ... a /_ ) if and only if there is a /i G F* such that « , •••, < _ 2 ) =/i(au • •, a n _ 2 ). Proof: Clearly, M(a,p) and ^(013,-.om) a r e maximal sets containing both 0 and E\2 — E2\. Let M be a maximal set containing 0 and E\2 — E2i, then by the proof of Proposition 4.8 M. is either of the form 0 -x\2

#12

••• xin \

' ( ' , )

I

( % )

Z12, • • •, xln G F

V -Xin

or of the form

X'Q)

\

0

#12

#13

-^12

0

^23

-a?i3

-£23

0

( ' « )

#12, #13, ^23 G -F

0/

where P G GL2(F), detP = 1, and Q G GLn.2(F). For the first case let the first row of P be (a, /?), then (a, /?) 7^ 0 and jVf = X( a i J 0 ). For the second case let the first row of Q be (ai, • • •, a n -2) 5 then (ai, • • •, a n _ 2 ) ^ 0 and M = A* (ai ,..., an _ 2 ). The second statement of the corollary is clear. D Now let us study the intersection of maximal sets. Proposition 4.11: The intersection of all the maximal sets which contain two adjacent points in common can be carried under a transformation of the form (4.1) to £ = {x{E12-E21)\xeF}. (4.13) Proof: By Proposition 4.3 we can assume that the two adjacent points are carried to 0 and Ei2 — E2\ under a transformation of the form (4.1). Let C be (4.13). Clearly £ C M{atP) for all (a,/?) G F™ but (a,/3) ^ 0,

4.2. Maximal Sets

165

and C C ^V(ai,...)an_2) for all (a 1? • • • ,a n _ 2 ) G F ^ " 2 ) but (
(°

Z12

0

-X12

0

Z13

0

—2>13

M{o,i) = <

0 \

> ^ 1 2 , ^ 1 3 , ••

s

•, *i» € F

0 ^I

0

/

—«ln

j

and Ai(i,o)nM(o,i) = C. Hence C is the intersection of all the maximal sets containing 0 and E12 — E21 in common.

□ Definition 4.4: The intersection of all the maximal sets containing two adjacent points in common is called a line. □ Proposition 4.11 can be restated as Corollary 4.12:

Through any two adjacent points there passes one and

only one line. A line can be carried under a transformation of the form (4.1) to (4.13).

□

Corollary 4.13:

The parametric equation of a line is

{ I'

where P G GLn(F)

H

0

X

\

—X

0

\P + K0 xeF\,

1

0/

and K0 G ICn(F). The parametric equation of a line in

the maximal set (4.5) is -

{

(°

1-

a«12 i2

*••

flln\ ain\

-012

+

0

I

(°

/

0 —612 b\2

612

*••

0 \-bln

bin\ 1 hn\

>

s

1 1

J

Chapter 4. Geometry of Alternate

166

Matrices

where (ai2, • • •, ai n ) =^ 0, and the parametric equation of a line in the maxi­ mal set (4.6) is /

0

a 12

a13

— CL\2 —
0 — d23

#23 0

\

/

+

0

&i2

b13

-bi2

0

623

-^13

— &23

0

V

\

*€F

0/

where (012,013,023) 7^ 0.

Q

Proposition 4.14: The intersection of two maximal sets containing two adjacent points in common can be carried under a transformation of the form (4.1) to (4.13) or (

V =

0

Z12

x13

-X12

0

0

-JTis

0

0

\ #12, Z13 € F

(4.14)

0/

I\

Proof: Let Mi and .M2 be two maximal sets containing two adjacent points in common. By Proposition 4.3 we can assume that the two adjacent points contained in both M\ and M2 are carried to 0 and E\2 — E2i under a transformation of the form (4.1). By Proposition 4.10 each one of Mi and M2 is either of the form M(a,p) or of the form A r ( 0l? ..., an _ 2 ). There are three cases to be considered. (a) M\ — M(a,p) and M2 = M(a',p>). (a', /?') are not proportional. Clearly M(a^) (b) (fli» £.

Since M\ ^ M2, (OJ, /?) and D M(a',p') = £-

Mi = jV(ai,...,an_2) and M2 = N(a'1,~.,a'n-2)' S i n c e Mx ^ M2, ' •> «n-2) and (a^, • • •, a^_2) are not proportional. Clearly M\C\M2 =

(c) Mi = M(a>p) and M2 = •A/r(a1,...,an_2). There is an element P e with det P = 1 such that (<*,/?)/> = (1,0) and there is an element Q G GLn-2(F)

such that

(01, • • • , a B _ 2 ) g = ( l , 0 , • • • , 0 ) .

GL2(F)

4.2. Maximal Sets

167

Then the transformation

*~XP Mp ,)• which is of the form (4.1), leaves both 0 and Ei2 — E2\ invariant and carries M(a,0)

and •A/r(a31,...,an_2)

A4(if0) is (4.5), ^ ( i . c - . o )

mto is

-^(i,o) and .A/(i,of...,o)j respectively. Clearly,

(4.6), and

M(i,o)nJ\f(i ) 0 ,,o) = P .

Definition 4.5: If the intersection of two maximal sets containing two adjacent points in common is not a line, then it is called a plane. □ Corollary 4.15:

The parametric equation of a plane is / 0 x -x 0 tp -2/0

y 0 0

\

P + K0 x,yeF\, 0/

V

where P G GLn(F) and KQ G K,n{F). in the maximal set (4.5) is

The parametric equation of a plane

{xA + yB + C\x, 2 / G F } , where ( A =

0 —a12

a12

(

0>ln\

0

,etc,

0

I

V-Gln

bln\

-612

,B-.

0

b12

I

\ - 6 In

and (ai2, • • •, flin) and (&i2, • • •, &in) are linearly independent; and the para­ metric equation of a plane in the maximal set (4.6) is {xA1 + yB1 + C1\x,

y,€F},

where (

A1 =

\

0

a12

a13

—#12

0

C&23

—ai3

—a23

0

\

/ ,*1

o/

=

V

0

612

6i3

— 612

0

&23

— &13

—^23

0

,etc, 0/

Chapter 4. Geometry of Alternate

168

Matrices

and (ai2, ai 3 , a2z) and (612,613,623) are linearly independent. P r o p o s i t i o n 4.16:

Denote the maximal (4.11) by M, then the map M

I

0

□

—► AG{n - 1, F)

S12

*\n\

P + #o'—► (a?i2,-'-» *in)

0

is bijective and carries lines and planes in the maximal set M. to lines and planes in AG(n — 1, F ) , respectively. Denote the maximal set (4.12) by .Af, then the map —+ AG(3, F ) /

V

0

^12

^13

-x12 -a?i3

0 -^23

a;23 0

\

P + KQ 1—> (a;12, £13, £23) 0/

is bijective and carries lines and planes in the maximal set J\f to lines and planes in AG(3, F ) , respectively. Proof: It follows immediately from Definitions 4.4 and 4.5 and Corollaries 4.13 and 4.15. □ Therefore we can say that the maximal set (4.11) has an (n — l)-dimensional affine space structure over F and the maximal set (4.12) has a 3-dimensional affine space structure over F.

4.3

Proof of the Fundamental Theorem

Proof of Theorem 4.4: The second statement of the theorem can be easily verified. We prove only the first statement in the following. We proceed in steps. (i) Let A be a bijective map of /C n (F) to itself and assume that both A and A'1 preserve the adjacency in ICn(F). Then A carries maximal sets into maximal sets. After subjecting A to the transformation X 1—> X - .4(0),

4.3. Proof of the Fundamental

Theorem

169

which is of the form (4.1), we can assume that .4(0) = 0.

(4.15)

(ii) Consider the image of the maximal set .M^o) (cf. (4.5)) under A. *4(-M(i)())) is also a maximal set. By Proposition 4.8, after subjecting A to a transformation of the form (4.1) we can assume that A{M(1}0))

(4.16)

= ^(1,0)

or ^ ( ^ ( i f 0 ) ) = ^(1,0,..,o).

(4.17)

If (4.17) occurs, then by Proposition 4.16 A induces a bijective map from AG(n — 1, F) to AG(3, F) carrying lines and planes into lines and planes, respectively. By the fundamental theorem of the affine geometry (Theorem 2.7) we must have n — 1 = 3 and, hence, n = 4. After subjecting A to the bijective map (4.4), which leaves 0 invariant, we also have (4.16). Let /

0

X\2

I

Xln\

0

^12

*1»\

^12

— 312

I

/

\-*ln

\—Xln

Then by Proposition 4.16 and the fundamental theorem of the affine geom­ etry we have [x\2, • • • , x j j = (Xi2, • • • , Xin)aP

+ (ai2, • • • , Cli n ),

where a is an automorphism of F , P G GLn-i(F), and (#12, • • •, ain) € _jr(w-i)# From (4.15) we deduce (ai2, • • •, a\n) = 0. Thus after subjecting A to the transformation X

1

V ,-M ,-)

which is of the form (4.2), we can assume that A leaves every element of ^(1,0) fixed.

Chapter 4. Geometry of Alternate

170 (iii)

Matrices

For i = 1, 2, • • •, n, let

r/ Mi =

\1

Xu

~*^lt

* * *

*^t—l,t

U

^t,t+l

''*

*^i\

I \ -a.n / ) where xu, £2t-, • • •, ^t-i,t, #t,t+i> * • *, #m € F . Then all A41, .M2, * * •, Mn are maximal sets, M.\ = .M(i,o)> a n d -^2 = ^ ( o , i ) - Clearly, M.\ C\ M.2 = {x(Ei2 — E2i)\x e F} = C. M2 contains £ , so does A(M2)- By Corollary 4.10 and the proof of Proposition 4.14 we must have A(M.2) = -M(a,p) f° r some a ^ G f and (a, /?) ^ 0. Since M\ ^ M2 we have A(M2) ^ M\. Therefore ft ^ 0. Then the transformation */

1 -a/?"1

\

/

1

1 -a/?"1

\ 1 (4.18)

X

1

\

1/

V

1/

leaves every element of A^i fixed and carries A(M>2) into .M2. Thus after subjecting A to the transformation (4.18), we can assume that A leaves Mi fixed elementwise and carries M2 into M.2Clearly, Mx D M{ = {x(Eu - E{1)\x G F } , i = 3, 4, • • •, n. Let & = M i H M i . Then A(M!)nA(Mi) = 4 ( A ) . Since (4.16) holds and A C A*i, we have M.\ C\ A(Aii) = Ci. In the same way as we proved Corollary 4.10 and Proposition 4.14 we can prove that A(Mi) consists of all the points of the form /

0

aiX12

(XiXij-i

—OiiXu

-OSiXij-i

-xu

Xu

ft«ifi-i

0

-QiiXln

•••

-PiXin

OtiXin \

0 Axi,,+i

•••

fcxu :

\

at-Xiff-+i

-fcxu ':

0

fcxu

4.3. Proof of the Fundamental

Theorem

171

where (x12, x13, • • •, a l n ) runs through F( n _ 1 ), a;, # G F , and # 7^ 0. Then after subjecting ,4 successively to the transformations ^ ^ - >V' ( /- x>-

^Eil)X{I OiP^EMl ai

- ctip^Ea), a^En),

i= = 3, 3, 4, • • ■, n, • »

" l

which leave .Mi fixed elementwise, we can assume that A leaves all Mi (i = 2, 3, • • •, n) fixed. Hence for each z = 2, 3, • • •, n, we can assume that t-l

n

X A(£xji(Eji-E (EjiEji)) K )'(■£«'.?• ~- Eji)) ij)+ Eij) + J2 Xij(Eij

j=i+l t-i

E - En) + E ^(Eij -Eji). = **(**■ * - ^ ) + E ^ - ( ^ - **•)■ = EE*U 3=1

i=i

Then .4 induces a transformation

An

AG{n-l,F)

j=M-i

-^ ,4G(n -- 1 , F )

X X AG{n-l,F) V^lt? An? «^t—l,n *^t,t+l5 J ^ m j ' - ( *—*AG{n-l,F) * 5 ^trj? t,t*+1? i , - , - - ' » i-l,ii \X\i) • • • , X t _i ) t ", X t)t _|_i, • • • , Xin) I > ^ i , - , ' ' ' j ^t—l,t» ^tjt+l? ' " ' ' Xin)i

which carries lines into lines and planes into planes. By the fundamental theorem of the affine geometry we can assume that {

X P*Et—l,n ^t,t'+l? *^t,t+l? ' ' *• 5 #xV^lt' t n j Y-*t5 * * » )= = l ^( l«t lj t ,' *' '' i5xi-l,ii t . t + l * * ' '* >> in) ( * « , •'•' ' '5 ^ t - l . t ? x^iti+li

where bti> Pi2, p«,t-i>Pt\«+1> Pt,«+i> ' •'

Since A leaves Mi fixed elementwise, pn = l(i = 2,3, • • •, n) and cr2 = cr3 = . . . = crn — 1. Also, for i > 1, since the points n

t-l

Eij) + xxu(Eii J2 ( F i , - ---Ej,) F.x) En) + £J2 x^
i=t+i i=«'+i

and tt -- ll

n n

J Z ^ ' ( ^ i * -- #tj) •O ^o) + + $E^ Xij(Eij *«(£«■ --■^Eji) J3=1 =I

i=»+l >=«+!

Chapter 4. Geometry of Alternate

172

Matrices

are adjacent, p{2 = Pi3 = • • • = p%,i-i = Pi,i+i = '" = Pin = A;. Considering the image of the intersection of Mi and Mj under A, we get immediately A2 = A3 = • • • = An = A. After subjecting A to the transformation l M

A-1/*"-1*) X (

H

A"1/*""1)) '

we can assume that A leaves each point in Ai{ fixed, z = 1, 2, • • •, rc. (iv) Denote .A/(i,o,-,o) simply by Afi. By the proof of Proposition 4.8 there are only two maximal sets, namely M\ and .A/i, containing the following three points

/o 0, '

-10

V

1

f °0

\ o(- 2 y

and

-1

0 1 0 0 0 0

\

\ 0 (n-3)

Q(n-3) yJ

Since M\ is invariant under .4, A/i must be mapped to itself under A. Let

4

( ~ #01 2 ~#13

^12

#13

0

#23

~#23

0

V

( °*

\

#12 ~#13

#12

#13

0

#23

""#23

0

V

0/

\

O)

Then the map (X12, X13, £23) •

> (#12? #13> #23)

is a transformation of the 3-dimensional affine space which carries lines into lines and planes into planes. By the fundamental theorem of the affine geometry and A(0) = 0 we have (#12> #135 #23) = 0*12, #13, # 2 3 ) ^ ,

where a is an automorphism of F and P G GLz(F). Because points in •Mi, A^2, and M3 are fixed by 4 , it is easy to see that a = 1 and P = I. Therefore

A

(°

— »12

#12

#13

0

#23

-#23

0

\

( =

0)

0

#12

#13

~#12

0

^23

~#13

~#23

0

\

0/

4.3. Proof of the Fundamental Theorem

173

for a l l £12, £13, Z23 G F.

(v) In the following we prove by induction on r that each element of the form £

xik(Eik-Eki),

(4.19)

l
where Xik G F (1 < i < k < r), is fixed by A Let r = n, then our theorem will be proved. The case r = 3 has already been proved in Step (iv). Now let r > 4 and assume that each element of the form y^

Xik{Eik — Eki)

l
is fixed by ^4. We first show that each element of the form Xj =

£

xik{Eik

- E^) + xjr(Ejr

- Erj), 1 < j < r - 1,

l
is left fixed by A. Assume that

A(X1)=

£ *rfc(£«-E«). l
Since Xi is adjacent with £

*,■*(£* - £ « ) + \{E12 - E21)

for every A G F ,

l
^4(Xi) is also adjacent with them. It follows that all 3 x 3 minors of £

x*ik(Eik - EM) -

l
£

*ik{Eik - En) - \(E!2 - E21)

l
is equal to zero for all A G F . Consequently, all 2 x 2 minors of X) 3
x

*ik(Eik — Eki) —

Yl

Xik(Eik — Eki)

3
is equal to zero. Thus the above matrix is of rank < 1. But it is an alternate matrix, therefore it is the zero matrix. It follows that x*^ — Xik for 3 < % <

174

Chapter 4. Geometry of Alternate

Matrices

k < r - 1 and x^ = 0 for j < /, 3 < j < n - 1, r < I < n. Xi is also adjacent with £

xik{Eik - E^) + A(£ 13 - £31)

for every A G F

Xik{Eik - Eki) + A(£ 14 - EA1)

for every A G F,

l
and ]T l
from which we deduce in a similar way x ^ = x2k for 3 < A; < r — 1 and a?2/ = 0 for r < I < n. The arithmetic distance between X\ and y ^ xik{Eik — Eki) 2
is equal to -rank

^

*

2
xik(Eik -

Eki),

from which it follows that x\r = £i r and x^ = 0 for r < 1 < n. Therefore r-l

A*i)

x

ii(E"

= E

~ E*) + x*r{Eir - Erl) +

1=2

Yl

x

^(Eik

- Eki).

2
In a similar way, for j = 2,3, • • •, r, we can prove that A(Xj)

= £ x%(Eu - En) + £ +

Y,

x^En xik(Eik

- Eti) + xjr(Ejr- Eki).

Then the adjacency of X\ and J2

xik{Eik - Eki) + A(E 2r - Er2)

for every A G F

l
implies x\k = x\k for 3 < k < r — 1, and the adjacency of X\ and 22

xik{Eik - Eki) + A(E 3r - Er3)

l
implies x\2 = x 12 . Therefore A(X1) = X1.

for every A G -F

Erj)

4.3. Proof of the Fundamental Theorem

175

In a similar way we can prove that ) =j, Xj, A(Xjj)=X

j = 2,3,-2 , 3 , . . •,r. -,r.

(vi) Let x

^hh iij'2

= —

x

52 5Z

Eki) + + xXj (Ejjirir ik(Eik -- Eki) ik{Eik ■ jirir(E

Erjl + xXj r(Ej -- E - h/ErjrJ2 rjl)) + 2r 2J,), hr2(E J2r

l
1 < i i < J2 h < < r.

We are going to prove that A leaves each X J U 2 fixed. We prove only the case ji = 1 and j 2 — 2, and the other cases can be proved in a similar way. Assume that A(X12) =

E

***(£* --E^.

l
From the adjacency of X\2 and J2 52

x

E22i)i) Eki) + A(JE?i2 A(£i2 --- F Xik(E ik(Eik ik --- E^)

for every A € G F, F,

l
we deduce x*k = Xik for 3 < i < k < r — 1 and x^ = 0 for j < /, 3 < j < n — 1, r < I < n. From the adjacency of Xi2 and x

E 52

F r i ) for for every every A AG G FF,, Xik(Eik ik(Eik --- E Ffc + A(£ A(F 23 F32 + x£lri(E F i r -- Eri) 2 3 --- E ki)t) + 3 2)) + r ( lr

l < t < f c < r - l1 tVfcO-

we deduce xjfc = xijt for 4 < k < r and x^ = 0 for r + 1 < / < n. From the adjacency of Xi2 and x

F31) + xx2r (E2r2r Fkiw)) + + A(jBi3 A(F13 --- E ik(Eik 3tt(F "- E 2r{E 31) + t - fc -

^52

Er2) --- E

for every every A AG G FF,, for

lL<«
we deduce x ^ = #2fc for 4 < A; < r and x^ = 0 for r + 1 < I < n. Then the adjacency of X12 and

E 52

xik(Eik Xik(Eik

F rjr i)) Eki) + X{E A ( Fjrj r ■- E --- Eki)

F , j = 1,2,3 for every A G F,

l
implies x j 2 = #12> x\3 = £ 13 , £23 = £23. Therefore 2 ..A(X * ( *12 «)) = -- ^X112

176

Chapter 4. Geometry of Alternate

Matrices

(vii) Let X

Jihh =

^2

X

ik{Eik - Eki) +

l
5Z

X

Jr(Ejr -

Erj),

J=Jl,J2,J3

1 < j i < to < ja < r. We are going to prove that A leaves each XjYj2^ fixed. We prove only the case (ji, j2i j3) = (1, 2, 3), and the other cases can be proved in a similar way. Assume that

A(X123)=

J2

x'ik{Eik-Eki).

l
Then from the adjacency of X123 and £

xik{Eik - Eki) + A(£i2 - E21) + x2r(E2r - Er2) + x3r(E3r - Er3)

l
for every A £ F , we deduce x*k — Xik for 3 < i < k < r — 1, x%r = x 3 r , and x){ = 0

for

j < Z, 3 < j < n - 1, r < 1 < n, but (j, /) ^ (3, r ) .

From the adjacency of Xi23 and X)

*,■*(£<* - Eki) + A(^ 2 3 - £32) + xlr(Elr

- Eri) + x3r(E3r

- Er3)

l
for every A £ F , we deduce a:^ = a?!* for 4 < fc < r and x^ = 0 for r + 1 < 1 < n. From the adjacency of X123 and ^

**(£,■* - £ « ) + X(E13 - E31) + x2r(E2r - Er2) + x3r(E3r - Er3)

l
for every A £ F , we deduce x^k = x2k for 4 < k < r and x\x = 0 for r + 1 < I < n. Then the adjacency of X123 and J2

x

ik(Eik - Eki) + KEjr - Erj)

for every A £ F, j = 1,2,3,

l
implies x\2 = x 12 , # i 3 = a?i3, x ^ = #23- Therefore ^4(-^123) = -^123-

4.4. Application to Geometry

177

(viii) Proceeding in the above way or applying an inductive argument, finally we will achieve the conclusion that each element of the form (4.19) is left fixed by A. Hence Theorem 4.4 is completely proved. □ Without essential difficulty Theorem 4.4 can be generalized as follows. Theorem 4.17: Let F and F' be fields of any characteristic, n and n' be integers > 4, and A be a bijective map from K,n(F) to /C n /(F'). Assume that both A and A'1 preserve the adjacency, i.e., for any A i , X2 G K,n(F), X\ and X2 are adjacent in fCn(F) if and only if A(Xi) and A(X2) are adjacent in /C n /(F'). Then n = n', F is isomorphism to F ' , and when n > 4, A is of form A(X)

= ptpX'P

+ K0

for all X G Kn(F\

(4.20)

where p G F'*, P G GLn(F'), a is an isomorphism from F to F ' , and K0 G K,n(F')', when n = 4, A is of the form A{X) = plP{X*yP

+ # o for all X G /C 4 (F),

(4.21)

where X \—> X* is either the identity map of K,±(F) or the map (4.4) of K,±(F). Conversely, any map from ICn(F) to tCn(F') of the form (4.20) or (4.21) is bijective and both the map and its inverse preserve the adjacency.□

4.4

Application to Geometry

At first we assume that F is a field of characteristic not two and n is an integer > 2. Let

,x

H A ?)•

<«■»>

which is a 2n x In nonsingular symmetric matrix of index n in its normal form. A 2n x 2n matrix T over F is called an orthogonal matrix of order 2n with respect to S if *TST = S. Clearly, 2n x 2n orthogonal matrices with respect to S are nonsingular and they form a group with respect to the matrix multiplication, called the

178

Chapter 4. Geometry of Alternate

Matrices

orthogonal group of degree 2ra with respect to the 2n x 2ra symmetric matrix S defined by (4.22) over F and denoted by 02n(F, S). It is easy to check that the following elements belong to 02n(F? S). 1° ( ^ 2° (* 3° (

T

t { A

K

j\ T

^ ^ where A where *K T

eGLn(F)-

=-K;

J, where J 2 = J is a diagonal matrix.

Let m be an integer, 0 < m < 2n, and P be an m-dimensional subspace of the 2n-dimensional row vector space F^2n\ Denote a matrix representation of the subspace P also by P, which is an m x 2n matrix of rank m whose m rows form a basis of the subspace P if m > 1 and is O^1'271) if m = 0. Let PL = {veF^2n)\vStP

= 0},

and call PL the dual subspace of the subspace P relative to S. Clearly, PLStP = 0, and by Proposition 1.29 d i m P 1 = 2n — m. It is easy to see that the map P —>PL from the set of subspaces of p( 2 n ) to itself has the following properties: for any two subspaces P and Q

pCQ=^p±DQ±; and for any subspace P d i m P + d i m P 1 = 2n and (p±)± =

p

If we regard an m-dimensional subspace P as an (m — l)-flat in PG(2n — 1, F ) , then P x is a (2n — m — l)-flat, called the dual flat of P relative to S. Therefore the map P —> P- 1 from the set of flats in PG(2n — 1, F) to itself is a polarity of PG(2n — 1, F) and is called the orthogonal polarity defined

4.4. Application to Geometry

179

by the 2n x 2n nonsingular symmetry matrix S (4.22), or simply the orthog­ onal polarity. If PL = P , then P is called self-dual and the corresponding subspace P is also called self-dual. Clearly (j( n ) 0 ^ ) and ( 0 ^ J
A subspace W is self-dual if and only if d i m W = n

Proof: Suppose that W is self-dual, i.e., WL = W. dimVl^"1- = 2n we deduce d i m W = n. From WLStW WStW = 0.

From d i m W + — 0 we deduce

Conversely, Suppose that dim W = n and WS*W = 0. From WS*W = 0 we deduce W C WL. From dim W+dim WL = 2n we deduce dimV^ 1 = n. Therefore WL = W, i.e., W is self-dual. □ An m-dimensional subspace P is called totally isotropic (with respect to S) if PS*P = 0. A totally isotropic subspace is called maximal totally isotropic if its dimension is equal to the maximum of the dimensions of totally isotropic subspaces. Maximal totally isotropic subspaces are of dimension n. Proposition 4.18 can also be stated as follows. Proposition 4.19: A subspace is self-dual if and only if it is a maximal totally isotropic subspace. □ Now let F be a field of characteristic two, n an integer > 1, and G={°

I(

Q),

(4.23)

which corresponds to the nondegenerate quadratic form (Xi, X2l ' ' ' , X2n)Gt(xU

X2, • • • , X2n)

= XlXn+1 + X2Xn+2

-\

\- XnX2n

in 2n indeterminates x 1? a;2, • • •, x2n over F with index n. n is also called the index of G. Let A and B be two m x m matrices over F. We write A = B, whenever A — B is an m x m alternate matrix. A 2n x 2n matrix T over F is called an orthogonal matrix with respect to G if l

TGT = G.

Chapter 4. Geometry of Alternate

180

Matrices

Clearly, 2n x 2ra orthogonal matrices with respect to the matrix G denned by (4.23) are nonsingular and they form a group with respect to the matrix multiplication, called the orthogonal group of degree 2n over F with respect to the matrix G denned by (4.23) and denoted by 02n(F, G). It is easy to check that the following elements belong to 02n(F, G). 1° ( A

( 3° (

T

f(yl

_1)),whereA€GI„(n

K \ j , where K is an n x n alternate matrix; ,

T

), where J 2 = J is a diagonal matrix.

An ra-dimensional subspace P of the row vector space F(2n> is called totally singular, if PGlP = 0. If P is a totally singular subspace of dimension m, then m < n. A maximal totally singular subspace is a totally singular subspace of maximal dimension. Maximal totally singular subspaces are of dimension n, and (1^ 0 ^ ) and ( 0 ^ 1^) are examples of maximal totally singular subspaces. We put S = G +*G, then S is also of the form (4.22) and lTST

(4.24)

= S for all T G 02n(F, G).

We agree that when F is of characteristic not two, totally isotropic subspaces are also called totally singular subspaces. From now on let F be a field of any characteristic. We introduce the nota­ tion 02n{F) to denote either the group 02n{F, S) or the group 02n(F,G), when the characteristic of F is not two or two, respectively. The following proposition can be readily proved. Proposition 4.20: Let F be a field of any characteristic. Let W be an n-dimensional subspace of F^2n\ Write a matrix representation W of the subspace W in the block form W = (X

F),

(4.25)

where both X and Y axe n x n matrices. Then W is a maximal totally

4.4. Application to Geometry

181

singular subspace if and only if XlY

is an n x n alternate matrix.

□

Let W be a maximal totally singular subspace. If in the block form (4.25) of W rank X = n, then W has a matrix representation (7(n)

x^Y)

(4.26)

where X~XY is an n x n alternate matrix. Thus the set of maximal totally singular subspaces is called the projective space ofnxn alternate matrices over F and is denoted by VKn{F). It is also called the dual polar space of type Dn. The maximal totally singular subspaces are called the points of VKn(F). Any matrix representation of a point W G VK,n(F) is called a homogeneous coordinate of the point. If the matrix X in the matrix representation (4.25) is of rank < ra, then the point W is called a point at infinity. If the X in (4.25) is of rank n, then W is called a finite point, it has (4.26) as one of its homogeneous coordinates, and the n x n alternate matrix X~XY is called its non-homogeneous coordinates. 02n{F) is a subgroup of GL2n(F).

We know that GL2n(F)

has an action on

p(2n).

FWxGL2n(F)—>FW ( ( X i , X 2 , • • • , X 2 n ) , T) I

► ( X i , X 2 , * * * , a; 2n )2 n

and an induced action on the set of subspaces of F^2n^ (cf. Section 1.3). Hence, restricting these actions of GL2n(F) to 02n(F) we obtain an action of 02n{F) on F<2n) and an action 02n(F) on the set of subspaces of F^2n\ It is easy to verify that for any W G VK,n(F) and T G 02n(F) we have also WT G VK,n(F). Hence 0 2 n ( ^ ) has an action on VK,n(F). We have Proposition 4.21: 02n(F) carries points of VK,n(F) to points of VICn(F). Moreover, 02n(F) acts transitively on VICn(F). Proof: Let us now prove the second statement. Clearly, (/ 0) is a point VKn{F). To show that 02n(F) acts transitively on VKn{F) it is enough to show that any point of VKn{F) can be carried to (/ 0). Let W = (X Y) be any point of VKn{F). If X is of rank n, then W = X(I X^Y), where X X~ Y is alternate, and the orthogonal matrix

/

-X'1Y\

182

Chapter 4. Geometry of Alternate

Matrices

carries W to (J 0). Suppose that r a n k X = r. By Proposition 1.17 there are P, Q G GLn(F)

such that

PXQ--

OJ " V 0 !)■ '*>-(? Then the orthogonal matrix ■ ■1 )

(Q carries W to

CO)" ) (*Q),x

«). 0^ «)• * - -((7 ( ( ? S) Wi =

where Fi = P y ^ Q ) - 1 . Write *i in the block form >i=

^12\ U21

where Yu is r x r and y 22 is (n - r) x (n - r). From Wi G VICn(F) we deduce that y u is alternate and Y2\ = 0. Then ranky 2 2 = n - r. The orthogonal matrix ,N

//M

f1 I \\

j(n-r)

/(') j(n-r) (n r)

/(«-'•) I //

/(r)

7 " carries W\ to W2 = (X 2 Y2), where

x2---" If/(r)0 X X2 -[0

\

Yi

A

Y22)

is of rank n, which is reduced to the previous case.

□

Our present purpose is to characterize the elements of the group of motions of V)Cn(F) with as few geometric invariants as possible. Parallel to Definition 3.8 we have Definition 4.6: The arithmetic distance between two points W\ and W2 of VKn(F), denoted by ad(W x , W 2 ), is defined to be r, if dim(Wi+W 2 ) = n+r. When r = 1, they are said to be adjacent. □ The following proposition gives an equivalent form of Definition 4.6.

4.4. Application to Geometry

183

Proposition 4.22: Let W\ and W2 be any two points of VKn{F).

Then

ad(Wi, W22) = == rankWiS*W rankW 1 S i W 23 . Proof: Clearly both ad(Wi, W2) and rankH / iS r t W 2 are invariant under 02n(F). By Proposition 4.21 we can assume that W\ = (I 0). Let W2 = (X Y). Then

?)-

dim(W == rank (* 1 Y dim(Wi1 + W W22)) = and W1StW2 = *r. Therefore *i(Wu

° ) == nn++ rrank a n k Fy

W2) = rank W1StW2.

□

Parallel to Proposition 3.29 - 3.31 we have Proposition 4.23: Let Wi, W2, and W3 be any three points of VtCn(F). Then 1° ad(Wi, W22)) > 0; ad(Wi, W32)) = 0 if and only if Wi Wx = == W w22.. ad(Wk, W W22)) = = ad(W2 , Wi). 2° ad(Wi, Wi). 3° ad(Wi, W2) ++ ad(W ad(W22,, W W33) >> ad(Wi, ad(W l5 W3 ).

□

Proposition 4.24: Let W\ and W2 be two points of arithmetic distance r in VtCn(F). Then they can be carried simultaneously to

(/ 0) (10)

and

/0W

^

I 0Q

0

0

/j(n-r) (n_r)

_ j Q(n Q("-r)■ • »

Q

0

.

) •

Proof: By Proposition 4.21 we can assume that W\ = (I 0). Let W2 = (X y ) , then rankF 2 = r. There exist P, Q G G £ n ( F ) such that

PYQ--

_ //(r)

o(»-

-

.

)

■

Then the orthogonal matrix

('W-1)

J

leaves W\ = (I 0) invariant and carries W2 to W3 = (X3 I3), where t X33 = PX X PXt(Q~ (Q~1)1)

and and

/ // iW -,

y^33 == (

oQ (( nB -_rr)))') .

Chapter 4. Geometry of Alternate

184

Matrices

Write X3 in the block form ( ^11

^12 \

V X2\

^22 /

where I n i s r x r and X22 is (n — r)x(n — r). Since X f Y is alternate, J n is alternate and X21 = 0. Thus rankX 2 2 = n - r, i.e., X22 is nonsingular. Then W3 has a matrix representation

//

-X12X£\W

Then the orthogonal matrix

I

_(XU

I

0

/M ON

\ i

-Xu \

I 0

1)

leaves W\ invariant and carries W± to /0W

0

v o /( n - r )

1^

o

0

\

o( n - r ))' □

Proposition 4.25: 02n(F) leaves the arithmetic distance between any pair of points invariant. Moreover, for any fixed r, 0 < r < n, the set of pairs of points of VK,n{F), which are of arithmetic distance r, forms an orbit under 02n{F). U Parallel to Definition 3.9, Proposition 3.32, and Corollary 3.33 we have Definition 4.7: Let W, W € VKn(F). When W ^ W, the distance 7 7 between W and W , denoted by d(W, W ), is defined to be the least positive integer r for which there is a sequence of r + 1 points Wo = W, Wi, • • •, Wr = W such that Wi and Wi+1 are adjacent, i = 0, 1, • • •, r -1. When W = W7, we define d(W, W) = 0. D Proposition 4.26:

For any two points W, W € VfCn(F) ad(W, W ) = d(W, W ) . D

4.4. Application to Geometry

185

Corollary 4.27: Let A be a bijective map from VK,n(F) to itself and assume that both A and A'1 preserve the adjacency in V)Cn(F), then A preserves the arithmetic distance in VtCn(F). □ When F is of characteristic not two, let S be the matrix (4.22), and when F is of characteristic two, let G be the matrix (4.23). We call a 2n x 2n matrix T over F a generalized orthogonal matrix with respect to S (or G), if 'TST^pS

(or'TGTEE/JG),

where p G F* and p is called the multiplicator of T. The set of all 2n x 2n generalized orthogonal matrices with respect to S (or G) over F also forms a group with respect to matrix multiplication, called the generalized orthogonal group of degree 2n with respect to S (or G) over F and denoted by G02n(F, S) (or G02n(F, G), respectively). We introduce the notation G02n(F) to denote either the group G02n(F, S) or the group G02n(F, G) when F is of characteristic not two or two, respectively. Every element T G G02n(F) defines a bijective transformation on VICn(F) W>—>WT, which preserves the arithmetic distance in VK,n(F). In particular, both the map and its inverse preserve the adjacency in VtCn(F). Conversely, we have the fundamental theorem of the projective geometry of alternate matrices over any field which reads as follows. Theorem 4.28: Let F be a field of any characteristic and n be an integer > 3. Let A be a bijective map of VKn(F) to itself and assume that both A and A'1 preserve the adjacency, i.e., for any Wi, W2 G VtCn(F), dim(Wi + W2) = n + 1 if and only if dim(,A(Wi) + A{W2)) = n + 1. Then A is of the form W ^ WaT for any W G VKn(F), (4.27) where a is an automorphism of F and T G G02n(F). Conversely, any map of the form (4.27) from V)Cn(F) to itself is bijective and both the map and its inverse map preserve the adjacency. To prove this theorem we need some preparation.

Chapter 4. Geometry of. Alternate

186

Matrices

Let W e VK,n(F) and write W == (Wx (wi W w22 -'•• ■•

Ww2n ), 2n),

where w\, w2, • • •, w2n a ^ e the 2n column vectors of W. Let s be an integer, 0 < s < n, z'i, z2, • • •, i 5 , j i , j2, • * *, in-s be a permutation of 1, 2, • • •, n and 1 < z'i < i 2 < • • • < is < ^, 1 < j i < 3i < • * * < jn-s < ^- Define £tit 2 -». = { ^ € 'P/Cn(^)|(^t 1 • * * wt-5 Wn+h • •' Wn+jn-s)

is

nonsingular}.

When s = 0, we denote K /C<£ = ={W{W€ € / P/C VKnn(F)|(tt; {F)\(wn+1 ■■ • • ww2n2n) ) isisnonsingular}. nonsingular}. <S> n+ i w nw+n+2 2 •■ L e m m a 4.29: For any point W € £; lt2 ...^, multiplying PF from the left by the inverse of (w tl wi2 ••• wis wn+jl wn+J2 ••• w n + i n _ s ), we obtain a matrix representation of W whose ii-th, Z2-th, • • •, i s -th, (n + ji)th, (n + j 2 )-th, • • •, (n + j n _ 5 )-th columns form the identity matrix 1^ and the remaining n columns can be arranged into an n x n alternate matrix Z = (wn+il

wn+i2 ■••• n+is •■ wwn+u

wh Wh

whh W

■••• ■■

w « >jn>_„a-). .)•

(4.28)

Moreover, the map "► >Cn(F)

^ - t i « 2 •••'»

w*--> z

is a bijection with the property that W i—> Z and W* i—> Z* imply ad(W,W*) = 2ad(Z,Z*). Proof: Consider the case (ii, i2, • ••, i s ) = ( 1 , 2 , • • •, 5) as an example. We have I Xl2 y X\ 0 Y)-( Y) -- Q ^ 22 Yn" j ( B°_.a," lj W =-W-{X W-( X (X Y)-{ j(n- ) ) Y V 0 2l x22 and 2 + 2\ \\ ( ^11 X\ -^12 + Y ^21 v t\s

*XY * - ( ' « •

x2222

-{o

l

)■ )

By Proposition 4.20 X Y is alternate. Thus both Y\\ and X22 are alternate, and X12 + *l2i = 0. Hence

z ==

Y\\Yn

(

\Y21

Xv X2: ; )

4.4. Application to Geometry

187

is an n x n alternate matrix. For the proof of the second statement let W be as above and //M

X* Y& *1*22 *?1

Vo

*2*2 x*22

0 \\ 0 (n S ^ l / Y2\ / ( »"- )V; •'

By Proposition 4.22 we have t ad(W, tV*) = = ra,nkWS mnkWS'W* a,d(W,W*) W*

= rank 1( vv \v r* 22i1

v* v —

— ^21 12i

j

-*22 A 2 2 ■"— A^22/ 2 2 /

= rank(Z-Z*). D

The n x n alternate matrix Z in (4.28) is called the non-homogeneous coordinate of the point W G /Ctl;2...ts. L e m m a 4.30:

(J VKn{F)= (J

(J U

£,,,,.,,. ^ l;l i 2 ... ta .

0<s
Proof: Let VT be any point of VKn{F) and write W = (X y ) , where both X and y are n x n matrices. Assume that rank X = 5, then by Proposition 1.23 there is a Q G GLn(F) and an n x n permutation matrix P such that

\■ QXP=( * l2) w-ft" *■x0■ >/' X 1

where Xu is an s x s invertible matrix. Let

QY\P~ matrix. where I n is an 5 x 5 invertible Let -1)-

Yu\ Y„J'

then the transformation then the transformation

W^

w(p

\p- . . )

carries W = (X Y) to carries W = (X W Y)1to = (X1

W-(Y

*i)=(*n V\-fXn

X12

0 Xl2

Yu

Yl2

\

Chapter 4. Geometry of Alternate

188

Matrices

l Since Xx *YX is alternate, XX1 *Y21+X12 lY22 = 0. Thus fY21 = -X^X12 Y22. But (y 2 i Y22) is of rank n — s, hence rankF 2 2 = n — s. Therefore W\ G /Ci2...s. We can assume that P _ 1 carries 1, 2, • • •, s into z\, z2, • • •, zs. Then

L e m m a 4.31: Let Wi, W2, and W3 be three points of VJCn(F) for which ad(Wi, W2) - n, ad(Wi, W3) = rz - 1, and ad(W 2 , W3) = 1. Then they can be carried simultaneously by an orthogonal matrix to (7 ( n ) 0^n)), (0 ( n ) 7 ( n ) ), and 00 (1 \

c ^

respectively.

-1) o(»Q(n-l)

- . , ^) . ^ J(n-l)

/ ( " ■

Proof: By Proposition 4.24 we can assume that W\ = (1^ 0^) and W2 = (0 / W ) . Let W3 = (X Y), where both X and Y are rz x n matrices. By Proposition 4.22, f rTankX a n k * ■= rank W WW W33SS% ad(W22,, W W33)) = 1 2 2 = ad(W

and W3StW1 = ad(Wi, VK W33) = rz rrank a n kY y = rank WaS'Wi n --- 11.. By Proposition 1.19 there exist P, Q G GLn(F)

p y g =■(°

j(n- -

.

)

such that ■

The orthogonal transformation

^r">,)

W i-

leaves both W\ and W2 fixed and carries W3 to 0

1

w 4 == M S ( ' « - - )

« ) ■ = (*4

j(n- - » ) .

where X 4 = P X ^ Q " 1 ) . Since rankXi = 1 and rank W4 = n, the first row of X4 is nonzero and the other rows of X4 are scalar multiples of the first row. Thus there exists a matrix representation of VT4, denoted by W4 again, of the form Xl1 Xl2 x12 '" • • • Xln rri n °0 ur - (( XU "\ "l W, = ^4 -

^

g(n-l,n)

0(n-lfn)

j(n-l) ) ■

4.4. Application to Geometry Since W4 € VKn(F), XU

X12 Q(n-l,n)

189

by Proposition 4.20

/(»-1))-(

") (

is alternate. Therefore xi 2 tat ion of the form

0

X12

#13 g(n-l,n)

* * '

#1

•)

* = #in = 0 and VF4 has a matrix represen­

ts - (

0 ( n _i)

jr(n-l) J D

Parallel to Lemma 3.48 we have L e m m a 4.32: Let n > 4, A be a bijective map from ^ ( F ) to itself, and assume that both A and A'1 preserve the adjacency. Let (z'o, io) be a fixed pair of indices, where 1 < io < jo < n. Assume that A leaves every point whose (io, jo)-entry is nonzero fixed, then A leaves every point of ICn(F) fixed. Proof: We discuss only the pair of indices (1, 2), for the other cases can be treated in a similar way. Suppose that A keeps all points of Kn{F) whose (1, 2)-entries are nonzero fixed. Let

/0

0 0

X =

#13

*•

#23

*•

0

*

*

%2n

%n—l,n

o )

\

be a point of fCn(F) whose (1, 2)-entry is zero. By hypothesis, we can assume

that /0

A(X) =

0

x*13

U

x23

■■■

x\n

\

u

2n

o '-n-l.n

0

/

190

Chapter 4. Geometry of Alternate

Matrices

Clearly, X is adjacent to /0

1

A

0

£23 0

X14

X\n

X

X2n

24 X34

*

\

XZn

for all A G F.

0 %n—1, n

V

0

1

So A(X) is also adjacent to it. It follows that x%j = x2j, 4 < j < n and x*- = XJJ, 4 < z < j < n. Since X is adjacent to /0

1

X13

A

£i5

0

X23

^24

#25

0

•••

X2n

for all A G F , %n—l,n

0

V

/

we have #23 = £23 and x$j = x3j for j = 5, • • •, n. Consequently x%4 = £34. Since X is adjacent to

/o 1

Z13

*

Xln

0

A

•

X2n

0

*

*

\

for all A € F, %n—l,n

0

)

we have x^- = xij for j = 4, 5, • • •, n. Similarly, we have also x\3 = X13. Hence A{X) = X. □ We are now ready to prove the fundamental theorem of the projective ge­ ometry of alternate matrices over any field. Proof of Theorem 4.28: Let A be a bijective map from VlCn(F) to itself, where n > 3, and assume that both A and A'1 preserve the adjacency. By Corollary 4.27 A preserves the arithmetic distance. Consider first the case n > 5. We proceed in steps.

4.4. Application to Geometry

191

(i) (I 0) and (0 I) are two points with ad((7 0), (0 / ) ) = n, so a d ( 4 ( J 0), 4 ( 0 / ) ) = n. By Proposition 4.24, after subjecting A to an orthogonal transformation we can assume that A(I

0) = (/ 0) and ,4(0 I) = (0 I).

Let W = {X Y) G VK,n(F). r a n k y = n. But

By definition of £*, W G K+ if and only if

r a n k F = a d p 7 ) , (/ 0)). Therefore (X Y) G / Q if and only if ad((X F ) , (J 0)) = n. Since 4 ( 7 0) = (7 0), we have 4(/C^) = / Q . For each W G /C^ choose a homogeneous coordinate of W to be W = (X / ) , where X G /C n (F), and assume that 4 ( W ) = (A*(X) I). By Lemma 4.29, A* is a bijective map from lCn(F) to itself and preserves the arithmetic distance between any pair of points. Now n > 5, so by the fundamental theorem of alternate matrices A\X)

= p%PX"P + K0 for all X G /C n (F),

where a is an automorphism of F, p G i71*, P G GLn(F), and if0 € /Cn(jP). Since 4*(0) = 0, we have i^o = 0. Therefore after subjecting A to the following transformation

(XY)^((XY)(P~1P~1

tp))°

,

which is of the form (4.27), we can assume that A(X

I) = (X I) for all X G /C n (F),

i.e., A leaves every point of K^ fixed, and A(I

0) = (I 0).

(ii) Let ii, z2, • • •, zs, j i , J2, • * *, jn-s be a permutation of 1, 2, • • •, n such that 1 < z'i < z2 < • • • < is < n, 1 < j i < 32 < ''' < jn-s < n. Denote by Wi1i2...ia the point of V)Cn(F), which has a matrix representation whose z'l-th, i 2 -th, • • •, z>th, (n + ji)-th, (n + j 2 )-th, • • •, (n + j n _ s ) - t h columns are ei, e 2 , • • •, e s , e s +i, • • •, e n , respectively, and all other columns are 0's. Then Wilt-2...,-, G /Ctlt-2...,s. We assert that A leaves each W,lt-2...ta fixed. We

Chapter 4. Geometry of Alternate

192

Matrices

prove by induction on s. Clearly W^ = (0 7), so our assertion is true for 6 = 0. Now let s > 1 and assume that our assertion is true for s — 1. Let us prove that it is also true for s. It is enough to consider the case (zi, z2, • • •, zs) = (1, 2, • • •, s), for the other cases can be treated in a similar way. Let A(W1212.....S) =- (X (XY), A(W Y), ■ * )

=

where both X and Y are n x n matrices. By Proposition 4.22 we have / (0 7))=rankW 7)) = rankW I) = s ad(Wi 12...,5'(0 ad(W 123..„ ... s ,, (0 1 2 ... s S<(0 I) = s

and

(7 0)) 0)) == rank rank W W1212... ..*S'(I 0) = = n -n-s. s. ad(Wi ..,5 ,, (7 ad(W 12a... 0) SS\I Since A preserves the arithmetic distance,

rank(X Y)S\0 Y)Si(0 I)I) === sS ad((X ad((X Y), Y), (0 (0 //)) ) ) ===rank(X and

t ad((X =--r<mk(X (I 0) 0) === n71 --- 5s.. ad((X Y), F ) , (I (7 0)) 0)) = rank(X Y)S K)S*(J

That is, r a n k X = s and rank Y" = n — s. We can assume that

JT =

V0 (n_s ' •») j

and Y =

\ Y2 ' ) •

where I i is an 5 x n matrix of rank 5 and Y is an (n — s) x n matrix of rank n — 5. Write A"i = (xi (xi X1 =

where those of 1^.

• ■ •x nx)n) and x22 ••• and F 2Y2= =(t/i(t/iy2y2 • •• • • yn), y n ),

the column vectors of Xi and yu y2, • • •, yn are

Consider first the case 5 = 1. Then Xi is a 1 x n matrix of rank 1 and Y2 is an (n — 1) x n matrix of rank n — 1. W\ is adjacent to all the points (En — En 7 ^ ) for 1 < i < n. All these points lie in JC^ and we have assumed in Step (i) that A leaves every point in K+ fixed. So A(W{) = (X Y) is also adjacent to all of them. It follows that n >)) = 1 for ti = 2, 3, • ••,•, n,n, rank(X Y)S\E y)S"(£ JJ<(n) li lt - - Eaa

4.4. Application to Geometry

193

that is, rank r a n k (f * 1a \2/»

V yi

If x n / 0, n — 1 > 1. Xi ^ 0. By and (?/2 2/3

x2

•■

*» 0 •- ■•

x

o •••

Xi

"

0

x

*•'

o

»* 0

-2/1

-2/i

o

- ;; M „. * » ) == i1 for farti == 2,2 ,3, 3 , •- .• •,•, n. •

••• o y

then yx = y2 = - - - = yn-i — 0, which contradicts rank V = Therefore # n = 0. Similarly, x2 = x$ = • • • = x n _i = 0. Thus Proposition 4.20 X lY is alternate, from which we deduce yi = 0 • • • 2/n) is an (n — 1) x (n — 1) nonsingular matrix. Let p

=(Xl \ V then P is nonsingular and P - a ( X

2 / 2 2 / 3 - •• » „ )) .' V2V3 •" VnJ F ) = W1. Hence >t(Wi) = Wi.

Then consider the case s > 1. Clearly, Wi2...s is adjacent to Wi2...(s-i), W/i2..(s-2)5? • • * 5 and W23...*, which are all left fixed by A by induction hy­ pothesis. So A(Wi2...s)=1(X Y) is also adjacent to all of them. Thus 0 y,-i 2/ 5-i

xs xs+1 •• ° *• Y0 ••••• '" **) = *> ■ 0o ;y 0

rank(° 2/2 ° 2/2 ••• rank ( Vyi V2/1

0 i/ 2/ss_i -i

lj

x -i ° V 0

rankf° ° -

rank ( V2/1 ••• V2/1 2/2 2/2 *••

s

0 xx i ° ^0 -::: *»W y ••• *0 )/ - • • s+

s5

0 • • • 0 z 5 + i ••• rank ( * 0 V 0 2/2 2/3 ••• 2/s If a;n 7^ 0, then all xi, £2, * * * ? ^n-i are multiples of x n and rankXi = 1, which contradicts the assumption s > 1. Therefore xn = 0. Similarly, xs+i = x s + 2 = • • • = x n = 0. Thus

?)-»•

X

-{

0(ns))

where Xu is an s x 5 nonsingular matrix. By Proposition 4.20 X%Y is alternate. It follows that

(s) _ (° r=

1W'

where Y22 is an (ra — s) x (n — 3) nonsingular matrix. Let

/>=(*"

Y22)

194

Chapter 4. Geometry of Alternate

Matrices

then P is nonsingular and P *(X Y) = Wi2...s. Hence A(Wu...s) The induction proof is now completed.

= Wi2...s>

(iii) Let ii, z2, • • •, i s , j i , J2> • * *, jn-s be again a permutation of 1,2, • • •, n such that I < ii < i2 < — - < is < n and 1 < j i < j 2 < • • • < jn-s < ™- Let W G VKn(F). Clearly, W G /C^.-.-. if and only if ad(W, W jlJ2 ... jn _ s ) = n. By Step (ii) -4(W ilJ2 ... jn _ 5 ) = Whh...jn_,. Therefore A()Cili2...is) = /Ctlt-2...tV We prove by induction on s that A leaves each /C^...^ fixed elementwise. For s = 0, by Step (i) we know that A leaves tCfixedelementwise. Let s > 0 and assume that our assertion holds for s — 1. Without loss of generality we consider only the case (z'i, i 2 , • • •, is) = (1, 2, • • •, s). Let W G £i 2 ... 5 , then W has a matrix representation W = (X F ) , where X = (ci e2 • • • e5_i e s x s + i • • • xn),

Y = (y1y2 • • • ys e s +i • • • e n ).

Denote the entry at (i, j ) position in V by yij. If t/5,5_i 7^ 0 then W G ^12"S-2,s and by induction hypothesis >l(W) = W. Hence by Lemma 4.32 A leaves /Ci2...s fixed elementwise. (iv) By Lemma 4.30 we conclude that A leaves VK,n(F) fixed elementwise. We have completed the proof of Theorem 4.28 for the case n > 5. Now consider the case n = 4. We start with three points of VfCn(F), are Wk = (J<4> 0 « ) , W2 = (0 (4) / ( 4 ) ) , and W

3

=(

1 o(3)

°

/(3)

they

).

By Lemma 4.31, after subjecting A to an orthogonal transformation we can assume that A(Wi) = Wi for 2 = 1,2, 3. As in case n > 5, from ^(W^) = W\ we deduce A{K^) = JC we have A(W) = (A*(X)

I),

where A* is a bijective map from IC^F) to itself and preserves the arith­ metic distance between any pair of points. By the fundamental theorem of

4.4. Application to Geometry

195

alternate matrices we have A*(X) = ptPiX'YP

+ K0

for all X G £ 4 ( F ) ,

where o is an automorphism of F , p G i*1*, P G GL4(F), K0 G /C 4 (F), and X* = X for all X G /C 4 (F) or X i—♦ X* is the map (4.4). Since ^4*(0) = 0, we have K0 = 0. Therefore after subjecting .4 to the following transformation

(x y)—>((* y ) ( ' ~ i > ~

tp))

,

(4.29)

which is of the form (4.27), we can assume that A(X

I) = (X* I)

(4.30)

for all X € K4(F)

and A(I

0) = (/ 0).

However, after subjecting A to the transformation (4.29), we have

A(W3)=(^

'xu

aij2

z13

z14 ,)•

where IQ is a 3 x 4 matrix of rank 3. By Proposition 1.23 we can assume that Y0 has one of the following forms 1

2/i

1

2/2

1 1

Vi 1

i

\

2/1

\ 1

2/2

/o , and

l 1

1/

2/3,

where yi, 2/2? 2/3 £ ^ - Denote the above four matrices by Y\, Y2, Y3, and Y4 in succession. We prove that the first three cases cannot occur. At first, assume that Y0 = Y\. That is, /#H

£12

#13

£14

0

0

0

1

A(W3) = V

0 \

yi

1

yz)

Chapter 4. Geometry of Alternate

196

Matrices

By Proposition 4.20,

*/0 0 0

0\

#11 #12 Xis X14

0

(

0 xu + xi4yi

1 2/2 V i 2/3/ x12 + x14y2 x13 + x14y3 \ 0 )

is alternate. Thus Xn + xi4yi = a?i2 + £i42/2 = #13 + #142/3 = 0. Therefore x14 ^ 0 and we can assume that

f-yi

-2/2 -»3 i

A(W3) =

o o o 1

V

o\ 2/2

1 2/3/

Let W-fc = ( £ * - Eki / ) , 1 < i < k < 4. By (4.30) A(W12) = W12. By Proposition 4.22 ad(W 3 , W12) = rank W^SiW12 = 1 and ad(^(W 3 ), >l(Wi2)) = rank>l(Wr3)S,*Wi2 = 3, which contradicts the assumption that A preserves the adjacency. Therefore the case Y0 = Y\ cannot occur. We can prove in a similar way that the cases YQ = Y2 and Y0 = Y3 also cannot occur. Therefore the only possible case is YQ = Y4. That is, we must have A{W3) = W3. Suppose that the map X \—> X*(X £ fC4(F)) is of the form (4.4), then A(W14) = W23. By Proposition 4.22, ad(W 3 , W14) = rank W3StW14 = 1

4.4. Application to Geometry

197

and *i{A(W3), A(W A(W1414)))) --= = rank =■■ 3, rankW W33SStW*Wn ad(,4(W3), 23 = which is a contradiction. Therefore the map X \—> X*(X identity map and (4.30) must be

G tC4(F)) is the

A(X /) A{X I) === (X X €G JC KA4{F). (F). ( * I) ' ) for all X Then, proceeding in the same way as the proof for the case n > 5, we will arrive at the conclusion A(W) = =■-W W for all W G VKA4(F). A{W) € -P/C (F). This completes the proof of the case n = 4. The proof of the case n = 3 is left to the reader.

□

Without essential difficulty Theorem 4.28 can be generalized as follows. Theorem 4.33: Let F and F' be fields of any characteristic, n and n' be integers > 3. Let A be a bijective map from VK,n(F) to V)Cn(F') and assume that both A and A'1 preserve the adjacency. Then n = n', F is isomorphic to F'y and A is of the form W Hi— W - -f ►1W°T PT

forall all W W €G V)C for VKn(F), (F),

(4.31)

where cr is an isomorphism from JF to F' and T G G0 2 n (^ 1 / )- Conversely, any map of the form (4.31) from VKn(F) to VKn{F') is bijective and both the map and its inverse map preserve the adjacency. □ More generally, let n be an integer, n > 1,

(°

/(»)

\

• J-

s = /(»)

when F is of characteristic not two,

(4.32)

when F is of characteristic two.

(4.33)

and

/0 G=

7(n)

( °

\

J'

Chapter 4. Geometry of Alternate

198

Matrices

In the first case A is a 8 x 8 symmetric matrix which is definite, i.e., xA tx = 0 for x e F^ implies x = 0. In the second case A is a 8 x 8 upper triangular matrix which is also definite. We define the orthogonal group and the generalized orthogonal group of degree 2n + 8 over F with respect to the matrix S (4.32) and G (4.33), when F is of characteristic not two and two, respectively, by 02n+s(F, G02n+s(F,

S) = {Te

GL2^s(F)\tTST

S) = {T€ Glv^sWTST

= S},

= pS, where p € F*},

and 02n+s(F, G02n+s(F,

G) = {T€ GLtn+siFtfTGT

G) = {Te GL^s(F)\*TGT

= G},

= pG, where p € F*}.

We introduce the notation 02n+s(F) to denote either the group 02n+6(F, S) or the group 02n+s(F, G) when F is of characteristic not two or two, respec­ tively. Similarly we introduce the notation G02n+s{F). An m-dimensional subspace P of the (2n + £)-dimensional row vector space if p$tp = 0 (or PG*P = 0). When F is of F(2n+s) i s c a l l e d totally si7lgUiar characteristic not two, totally singular subspaces are usually called totally isotropic subspaces. It follows immediately from Witt's Theorem that the maximal dimension of totally singular subspace is n. Totally singular subspaces of dimension n are called maximal totally singular subspaces. Denote by Nn the set of maximal totally singular subspaces, and call the subspaces in Nn the points. Clearly, when 8 = 0, Nn = VtCn(F), the dual polar space of type Dn. When 8 = 1, Nn is called the dual polar space of type Bn. When 8 = 2, Nn is called the dual polar space of type 2J9n+i. Define two points W\ and W2 in Nn to be adjacent, if dim(W/1 + W2) =n + l. Then we have the following generalization of Theorem 4.28. T h e o r e m 4.34: Let F be a field of any characteristic, n be an integer > 3, and S and G be the (2n + 8) x (2n + 8) matrix defined by (4.32) and (4.33), respectively. Let A be a bijective map from Nn to itself and assume that both A and A'1 preserve the adjacency. Then A is of the form W h—> WaT

for all

W G Nni

4.5. Application to Geometry

(Continued)

199

where cr is an automorphism of F and T G G02n+s(F). Conversely, any map of the above form from Nn to itself is bijective and both the map and its inverse map preserve the adjacency. For the proof of Theorem 4.34, cf. Chow 1949 and Dieudonne 1951, or Dieudonne 1955.

4.5

Application to Geometry (Continued)

In this section let F again be a field of any characterstic, n be any integer > 2, and / 0 S = ( T(n\

7(n)\ ) , when F is of characteristic not two, n

and G = (

wnen

n )'

F is of characteristic two.

In the preceding section we defined the orthogonal group of degree 2n over F with respect to S or G to be 0 2 n ( F , S) - {T € GL2n{F)\lTST

= S}

or GL2n(F)\%TGT

0 2 n ( F , G) = {Te

= G},

respectively, and we called the elements of 02n(F, S) and 02n(F, G) orthog­ onal matrices. Now we distinguish the cases when F is of characteristic not two or two. Consider first the case when F is of characteristic not two. It is easy to see that orthogonal matrices are of determinant ± 1 . An orthogonal matrix T is called a proper ox improper orthogonal matrixii det T = 1 or — 1, respectively. For instance, for any odd integer r, 1 < r < n, the matrix //(n-r)

o 0

o V

/

(r)

\ r

/(n-r)

7< )

0 )

(4.34)

Chapter 4. Geometry of Alternate

200

Matrices

is an improper orthogonal matrix. Clearly, the set of proper orthogonal matrices forms a subgroup of index 2 of 02n{F, 5 ) , which is called the proper orthogonal group of degree 2n over F with respect to S and denoted by

Otn(F, S). Then consider the case when F is of characteristic two. Let T £ 02n(F, G) and write

T-- A A T=(

B

) = ( c DJ'

where

B— A A == (atfc)i (a,ifc)iB = (bik)i
D(T) =

- l
bikCiki

which is called the Dickson invariant of T. It is known that D(T) — 0 or 1. For instance, the Dickson invariant of

A ({

1 '(A' ( A - 1)]' ))'

where A G GLn(F), is 0, and the Dickson invariant of the matrix (4.34) is equal to 0 or 1 according as r is even or odd. An orthogonal matrix T is called proper or improper, if D(T) = 0 or 1, respectively. It is known that the set of proper orthogonal matrices forms a subgroup of index 2 of 02n{F, G), which is called the proper orthogonal group and denoted by O ^ F , G). For the details about the Dickson invariant and the proper orthogonal group over a field of characteristic two, cf. Dickson 1900 or Dieudonne 1955. We use the notation O^F) to denote either the group O ^ F , S) or the group (?2n(F, G), when F is of characteristic not two or two, respectively. Proposition 4.35: The set of maximal totally singular subspaces of F^2n^ with respect to S is partitioned into two orbits under the group O^F). The following two maximal totally isotropic subspaces (n) ) a and (/<"> 0<X«>) (/<"> nd

fall into distinct orbits under

(n -1)

//(»- -i)

(^"^

O^F).

Q

0

°0(n "" 1)

x)

.)

4.5. Application to Geometry

(Continued)

201

Proof: By Proposition 4.21 the set of maximal totally isotropic subspaces of jp(2n) forms an orbit under 02n(F). O^F) is a subgroup of index 2 of 02n(F)- Thus if we can prove the second statement of the proposition, then the first statement follows immediately. Assume that there is an element T e Otn(F) such that

(/w o)r = W where P G GLn(F).

J

0

Write T in the block form T=(A

B

)

where A, 5 , C, i) are n x n matrices. Write A, 5 , C, Z), and P in block forms; for instance, V A2i A 2 2 J where A n is (n — 1) x (n — 1) and A22 is 1 x 1. Then

and

Thus

(An \A21

A12\ A22)

(B11 \B2i

512\ B22)

=

(Pn \P21 (0 V0

//(n-i)

0\ OJ P12\ P22)

\

0

1 J(n-l)

V

1

= (£

^)

(4-35)

0/

Since (4.35) is an orthogonal matrix, we have D* = t(P~1). It follows that (4.35) is a proper orthogonal matrix, and, hence, T is improper, which is a contradiction. Q Each orbit of maximal totally isotropic subspaces under 02n{F) is called an irreducible space of the projective space of n x n alternate matrices over F , VKn{F). The irreducible space containing (7 ( n ) 0 (n) ) will be denoted by VIC+(F) and that containing / j(n-l)

V

Q(n-1)

0

0

202 by

Chapter 4. Geometry of Alternate

Matrices

VK-{F).

P r o p o s i t i o n 4.36: points of VK+(F)

The arithmetic distance between any two distinct

(or VtC~(F)) is always even.

Proof: Let W\ and W2 be two points of VKn{F) which lie in the same orbit of Otn{F). That is, they belong simultaneously to VJC+(F) or P/C"(F). Let ad(Wi, W2) = r. By Proposition 4.24 there is an element 7\ G 02n(F) such that -r) 0 B . ^(I(n >) and - (/(") = (/'"> 0< 0'")) and W W2T! ^ T= ^~ ^ W11TT11-Q ° /(r)).

0

/('))'

It is easy to see that W\T and W2T belong to the same orbit under O^F). Proceeding as in the proof of Proposition 4.35, we can show that r is even.D Parallel to Proposition 4.24 and 4.25 we have P r o p o s i t i o n 4.37: Let W\ and W2 be two points of VK^{F) with ad(Wi, W2)=2r, where r is an integer > 0. Then they can be carried si­ multaneously under 02n(F) into / / ( » ■

-3r)

0

(n) (/(») and ( / ( - 2 r ) ( J W O0W )) and

Proof:

0

0

°

/ (( 2 rr )) )) -.

By Proposition 4.24 there is an element T G 02n{F) such that (n Wir == (/<») o ') WiT _

W2T =

/

/

= (

»

(J(n) Q(n)j

-2r)

f

0

■

/ r(n-2r)

0

Q

J<3'>)-

u

bince both Wx and ( JW ^ 2T0(n)) belong to the same irreducible space VJC+(F), we have T G Of n (F). □ Proposition 4.38: The set of pairs of points in VtC+(F) which are of the same arithmetic distance 2r, where 1 < r < [|], forms an orbit under

Otn(F).

O

Correspondingly, we have Proposition 4.39: Let W\ and W2 be two points of VtC~(F) with / ad(Wi, W 2 )=2r, where r is an integer > 0. Then they can be carried si-

4.5. Application to Geometry multaneously under 02n{F) 'j(n-l)

into

\ ^

Q(n-l)

203

(Continued)

/ J{n-2r-l) ^ ^jr( n _2r-l)

Q Q

\

0 or //(n-i)

o

o(n_1)

\

lj

and

/0(n_1)

{

i

7( n_1 )

o j ' if2r = "-

Proposition 4.40: The set of pairs of points in VK,~(F) which are of the same arithmetic distance 2r, where 1 < r < [|], form an orbit under

Otn(F).

□

Definition 4.8: Two points W\ and W2 belonging to the same irreducible space VtC+(F) (or VK,~(F)) are said to be adjacent, if ad(VFi, W2) = 2, i.e., if dim(Wi + W2) = n + 2. □ Definition 4.9: Let W and W be two points belonging to the same irreducible space VIC+{F) (or VtC^(F)). When W ^ W", the distance d(W, W) between W and W is defined to be the least positive integer r for which there are r + 1 points W0 = W, Wi, • • •, Wr = W such that W{ and Wi+i are adjacent, where z = 0, 1, • • •, r — 1. When VK = W7, define d(W, W) = 0. □ Proposition 4.41: Let VK and W be two points belonging to the same irreducible space VIC+(F) (or VK~{F)), then d(W, W ) = iad(W, W')D

Corollary 4.42: Let A be a bijective map from VK+{F) (or VK~{F)) to itself and assume that both A and *4 - 1 preserve the adjacency. Then A preserves also the arithmetic distance. □ We recall that a 2n x 2n matrix T over F is called a generalized orthogonal matrix with respect to the matrix S defined by (4.22) (or G defined by (4.23)) with multiplicator /), where p £ F*, if t

TST = PS

(ov'TGT^pG).

Chapter 4. Geometry of Alternate

204

Matrices

Clearly, for any p G F* the 2rc x 2n matrix T - (plin)

)

is generalized orthogonal with multiplicator p. If 7;, i = 1, 2, is generalized orthogonal with multiplicator />t, then T{T2 is generalized orthogonal with multiplicator pip2. Moreover, if T is generalized orthogonal with multipli­ cator /9, then TTp-i is an orthogonal matrix. If TTp-i is proper orthogonal, then T is called a proper generalized orthogonal matrix; otherwise, it is called an improper generalized orthogonal matrix. Clearly the set of all proper gen­ eralized orthogonal matrices forms a subgroup of index 2 of G02n(F), which is called the proper generalized orthogonal group and denoted by G O ^ F ) . It is clear that for any T G GOj n (F), T carries V*C+{F, 5) (and VtC~{F, S)) into itself and preserves the arithmetic distance. Conversely, we have the fundamental theorem of the irreducible space VJC+(F) (or VK,~(F)). Theorem 4.43: Let F be a field of any characteristic, n be an integer > 5, and A be a bijective map from the irreducible space V1C+(F) (or VK~(F)) to itself. Assume that both A and A~l preserves the adjacency. Then A is of the form W ^

WaT

for all W G VK$(F)

(or 7>/C"(F)),

(4.36)

where a is an automorphism of F and T G G?(9jn(F). Conversely, any map of VK+(F) (or V£-(F)) to itself of the form (4.36) is bijective for which both the map and its inverse map preserve the adjacency. Proof: We prove only the first statement. Since an improper orthogonal matrix carries VK+(F) into VK,~(F) and VK~(F) into P/C+(F) and pre­ serves the arithmetic distance, it is sufficient to prove the theorem for any one of the two irreducible spaces. Consider the case when n is even. To fix our ideas let us consider VK,+(F). We proceed in steps. (i) Let A be a bijective map from V1C+(F) to itself, and assume that both A and A'1 preserve the adjacency. By Corollary 4.42 A preserves the arithmetic distance. Since n is even, the point ( 0 ^ /( n )) belongs also to

4.5. Application to Geometry (Continued)

205

VIC+(F). Since ad((7 0), (0 /)) = n, we have a,d(A(I By Proposition 4.37 there is a T G 0* n (.F) such that

0), .4(0 7)) = n.

A(I 0)T == (7 A(I (7 0)0) and .4(0 I)T I)T ==- (0 /). (o /)• Thus after subjecting A to the bijective map

W^> W^WT WT for all W W€VJCt(F), G VK+(F), which is of the form (4.36), we can assume that A{I 0) ==--(io) A(I (7 0) and .4(0 == (0 A(0 7) I) -(o 7). /)• Let

n-

K+ =~-{(X VfC+(F)\r&nkY { ( * Y)Y)eeV£Z(F)\ra,nkY

==: nn}. }.

Clearly, (X Y) € K\ if and only if ad((A" Y), (7 0)) = n. Since .4(7 0) = (7 0), we have

•4(£J) -=-1C+. retAW We can choose the matrix representations of points in /Cj so that

n=

/Cj == {(X is3alternate}. alternate}. {(X I)\X I)\XU Clearly, the map

AC„(F) £J —- /C„(F) ( X 7/ )) -^-+X X (X

*?-

is bijective with the property that ad((X / ) , (X* I)) = 2 ad(X, X*). Hence A induces a bijective map

A*: K.n(F) --» ^n(F) A* : K ICn(F) n{F) —+ such that A(X .4(X 7) I) ==(A*(X) M* W I)') and A* preserves the arithmetic distance. We have assumed n > 5. By the fundamental theorem of alternate matrices we have t .4"'(*) === p/>«PX*P A*(X) PX"P ++ TCo, K(h

Chapter 4. Geometry of Alternate

206

Matrices

where p G F*, P G GLn(F), a is an automorphism of F , and K0 £ fCn(F). Since A(0 I) = (0 / ) , we have KQ = 0. Therefore after subjecting A to the transformation

H^^C" 1

tp))

,

which is of the form (4.36), we can assume that A leaves each point of /Cj fixed and A(I 0) == (/ (I 0). (ii) We adopt the notation of the proof of Theorem 4.28. Let s = 2r be even, where 0 < r < | . Let ii, i 2 , • • *, is, j i , j 2 , • * •, jn-s be a permutation of 1, 2, • • •, n such that 1 < i\ < i2 < • • • < is < n and < ji < j 2 < • • • < j n _ s < n. Then W ^ 2 . ^ G P/C+(F). We assert that ^(W- lt - a ..^) = Wi^..*.. We prove by induction on r. When r = 0, then s = 0 and we have *4(0 / ) = (0 / ) , so our assertion is true for r = 0. Assume that our assertion is true for r — 1, we are going to prove that it is also true for r. It is enough to consider the case (ii, i 2 , • • •, is) = (1,2, • • •, s), since the other cases can be treated in a similar way. Let A(W12... ) = •4(Wl == (X 2..S•-)

Y),

where both X and Y are n x n matrices. As in the proof of Theorem 4.28 we have r a n k X = s and rankY = n — s. Then we can choose the matrix representation of (X Y) so that /g(»,» and Y --=0Y2 and y=( V r 2 ' ) )■ , where X\ is an s x n matrix of rank 5 and Y is (n — s) x n of rank n — s. Write Xx = ((x • •• xxnn)) and and YY22 = = (y (yxx yy22 •■•• • yyn),n ), Xi x xi xz22 • ■ Xl X * ==(-(0 ( » - i , » ) )

n)J

where those of Y2-

the column vectors of Xx and yu y2, • • •, yn are

Consider first the case r = 1, i.e., 5 = 2. From the adjacency of (/(2) W12 = (^ Wia= ^

0

0

0

\ n d (F lt - -- £F, til j/(»-') and ( n - 2 ) JJ a ( ^

(n) /JW ) for for 22 << ii < n, n,

4.5. Application to Geometry (Continued)

207

we deduce x2

-' • Z t - i " ' *" •■■■ 0 ■•

X£ ; + i

*

- '[ *XQ"n)U= 22 for for 22<< ii < nn.. •• • • 0 j ~ ~ (4.37) (4.37) We have assumed n > 5. It follows from (4.37) that x 3 = x4 = • • • = £ n = 0. Therefore rank r a n k (f * 1 \yi V2/t

*2 0

Xi

^

-- 2y/ i1

0

x* ==JXu (

2 Q(" -*>) o<">)

where Xu is a 2 x 2 nonsingular matrix. Write

Y=

\ ^21

^22 / ' Y22)'

Y21 is an (n — 2) x 2 matrix and Y22 is (n —2) x (n — 2) matrix. By Proposition 4.20 Xu ^Y22l\\\ XVt^r
- \ oVo

0

);

is alternate. It follows that Y2\ = 0 and Y22 is nonsingular. Thus

T , u u /(«-- , ) • (XY) = 0 *w F 2 J r o°,.-,)0 <*M*".°«,)-( "»j" =(

_

A n

(Ml

/i^'

Therefore -4(WM) - 4 ( W M ) === Wl2. W12.

Then consider the case r > 1, i.e., s > 2. By induction hypothesis •A(Wi == Wi 2)) = 4(W».2...(._ ..(.--2)) w 122......(..(j--22)) -. From the adjacency of W\2-.s rank r a n k ( U i V2/1

•

an(

l Wi2-(«-2)

0

we

deduce

Xg

2/2 • • yy»-2 5-2 2/2 ••••

0

0

0

•. .:: .

2 o")= o j - 2 --

Similarly, we have U • '• XsXs-2 r a n k(f ° ° ~2 rank V 2/1 2/2 * •■•• • 0 0 V2/1 2/2

U ° *x '$ *;0+ 1 ■-■;; X^ 0 =" ) 2= 2, , etc. 00 y _i 0 ••• 0J s 2/5-1

From these rank relations it follows that xs+i = xs+2 = • • • = xn = 0. Therefore

X-- = ( * »

o(»--•))>

Chapter 4. Geometry of Alternate

208

Matrices

where X\\ is an s x s nonsingular matrix. As the case 5 = 2 we also have _ y =

/QW ^22 > 1W'

where Y22 is an (n — s) x (n — s) nonsingular matrix. Therefore A(W ... . ^ ( W12w...-a) =--W w12 12...a.a ■*)

=

This completes the induction proof. (Hi) Let s = 2r be even and i\, i2, • • •, ia, ji, j 2 , ' " ■> jn-s be a permutation of 1, 2, • • •, n for which 1 < i\ < i2 < ■ ■ • < ia < n and 1 < j \ < j 2 < ■ • ■ < jn-s < n. For W € VK+{F), write W == (wi w22 -■■■ w " W2n 2n), where u>i, w2, • • •, w2n are the column vectors of W. Define is nonsingular}. ^th-i. =={W € VKt{F)\{Wil il ■■■ wisWn+h wn+jl ' •■• ■wn■+jw is nonsingular}. ICU~<. {WeVlCi(F)\(w •' ' ™is n_n+jn 3) _,) Then for any W G V)C+{F), W € K+h-i. i f a n d o n l v i f a d W whh-in-.) = n. By Step (ii) A(Whh...is) = Wilh...is. Therefore Afc+i,..*.) = ^ t v - V We assert that A leaves each K,fli2...iafixedelementwise. We apply induction on r. For r = 0, we have s = 0. By Step (i) we know that A leaves /Cj fixed elementwise. Let r > 0 and assume that our assertion holds for r — 1. For W G K,fli2...ia we choose a matrix representation of W whose ii-th, i 2 -th, • • •, i 5 -th, (n + ji)-th, (n + J2)-th, • • •, (n + j n - s ) - t h columns form the identity matrix I^n\ As in Lemma 4.29 we have a bijective map

!CU-.<.-+lC(F) ->/C(F) ^hh-i. W\—> (wnn+il ~* (W +h

w*-

w +i2 ■••• wn+is ■■ w Wnn+i2 n+is

whh w

wkJ2 ■••• w ••

wjs) «>j.)

which preserves the adjacency of pairs of points. Without loss of generality we consider only the case (i 1? z2, • • •, is) = (1, 2, • • ■, s). Let W G /C^..^, then W has a matrix representation W = (X F ) , where X = (e! e2 • • • es xs+1

xs+2

• • • £n)

4.5. Application to Geometry

(Continued)

209

and y = (yi 2/2 • • • Vs e s + i e s + 2 * • * en). Denote the (z, j)-entry of Y by y{j. If ys,s-\ ^ 0, then ys-X,s = -?/ s ,s-i 7^ 0. Thus VK G /Ci2...s-2- By induction hypothesis A(W) = W. Then by Lemma 4.32 A leaves K\2.s fixed elementwise. (iv)

Similar to Lemma 4.30, we have

VK+n{F)=

U

U

*&..*.,

o<s
□

Without essential difficulty Theorem 4.43 can be generalized as follows. T h e o r e m 4.44: Let F and F' be fields of any characteristic, n and n' be integers > 5. Let A be a bijective map from V)C+(F) (or VtC~(F)) to VK*,{F') (or VK,~,(F')) and assume that both A and A'1 preserve the adjacency. Then n = n', F is isomorphic to F'. Moreover, if A maps VK+(F) to VK,+{F') or maps VK~{F) to VK-(F'), then A is of the form W ►—► V T T for all W G P/C+(F) (or VK,~{F)),

(4.38)

where cr is an isomorphism from F to F ' and T G G02n(F'); if 4. maps VtC+(F) to VK-{F') or maps VtC~(F) to P/C+(F'), then .4 is of the form VT .—► W T

for all W G P/C+(F) (or VK,-{F)),

(4.39)

where cr is an isomorphism from F to F' and T is a 2n x 2n improper generalized orthogonal matrix over F'. Conversely, any map of the form (4.38) where T G GOtn(F) maps VK+{F) to VK+(F') and VK~(F) to VK,~(F') bijectively and both the map and its inverse preserve the adjacency of pair of points, and any map of the form (4.39), where T is an improper generalized orthogonal matrix maps V1C+(F) to VJC~(F') and VfC~(F) to V)C^(F') bijectively and both the map and its inverse preserve the adjacency of pairs of points. □

210

Chapter 4. Geometry of Alternate

Matrices

Remark 4.1: Both Theorems 4.43 and 4.44 are not true when n = 1, 2, or 3, for in these cases any pairs of points in VtC^(F) (or VK,~(F)) are adjacent. When n — 4, there is an exceptional bijective map from VK^(F) (or VK^{F)) to itself arising from the theory of triality such that both the map and its inverse preserve the adjacency and that it is not of the form (4.38). And we can prove that any bijective map from VK,%(F) (or V)C^(F)) to VK^{F') (or VK^{F')) such that both the map and its inverse preserve the adjacency is either of the form (4.38) or is the product of a map of the form (4.38) with the exceptional bijective map of VIC^^F') (or V1C^{F')^ respectively), and that any bijective map from V)C\{F) (or VK^{F)) to VK,±(F') (or VK^{F')) such that both the map and its inverse preserve the adjacency is either of the form (4.39) or is the product of a map of the form (4.39) with the exceptional map of VJC^(F/) (or VK^{F')^ respectively). For details, cf. Dieudonne 1955. More generally, when F is characteristic not two, for any T £ 02n+<$(F, S) T is called a proper or an improper orthogonal matrix if det T = 1 or —1, respectively. Clearly, the set of proper orthogonal matrices forms a subgroup of index 2 of 02n+s(F, 5), which is called the proper orthogonal group of de­ gree 2n + 6 over F with respect to S and denoted by O ^ + ^ F , &)• When F is of characteristic two, for any T £ 02n+6(F, G) the Dickson invariant D(T) can also be defined and takes the value 0 or 1. Those T £ 0 2 n+£(F, S) with D(T) = 0 are called proper orthogonal matrices and form a subgroup of index 2 of 02n+6{F, 5), called the proper orthogonal group of degree 2n + 6 over F with respect to G and denoted by 02n+s(^ G)- Otn+siF) G) is a subgroup of index 2 in 02n+s(F, G), and elements belonging to 02n+s(F, G)\02n+s(^ G) 1 are called improper orthogonal matrices. We use the notation O^n+^-f ) to denote either the group 0 2 f i/+5 (F, S) or the group O ^ n + s ^ G), when F is of characteristic not two or two, respectively. The set Nn of maximal to­ tally singular subspaces is also partitioned into two orbits under O ^ + ^ F ) , which are denoted by 7V+ and 7V~, and are called the irreducible spaces. It can be proved that the arithmetic distance between any two points of iV+ (or N~) is even. Define two points of N+ (or N~) to be adjacent if the arithmetic distance between them is 2. As in the case 8 = 0, generalized

4.6. Application to Graph Theory

211

orthogonal matrices can be defined and they form a group with respect to the matrix multiplication, which is called the generalized orthogonal group of degree 2n + 8 over F with respect to S (or G) and denoted by G02n+s{F, S) (or G02n+s{F) G)). Moreover, proper or improper generalized orthogonal matrices can also be defined and the former form the proper generalized orthogonal group of degree 2n + 8 over F with respect to S (or G) and is denoted by GO^+siF, S) (or GO^+siF, G)), which is a subgroup of index 2 of G02n+s(F, S) (or G0 2 n+£( J ^ G), respectively). We introduce similarly the notation G02n-{-8(F) and G O j n + 5 ( F ) . Then Theorem 4.43 can be gen­ eralized as follows. Theorem 4.45: Let F be a field of any characteristic, n be an integer > 5, S be the (2n + 8) x (2n + 8) matrix defined by (4.32), and G be defined by (4.33). Let A : N+ —> iV+ (or JV~ —> JV") be a bijective map and assume that both A and .A"1 preserve the adjacency. Then A is of the form W »—> WTT for all W G iV+ (or iV"),

(4.40)

where a is an automorphism of F and T 6 G02n+$(F). Conversely, any map of the form (4.40) from N+ (or N~) to itself is bijective and both the map and its inverse preserve the adjacency. □ For the proof, cf. Chow 1949 and Dieudonne 1951, or Dieudonne 1955.

4.6

Application to Graph Theory

Let F be a field of any characteristic and n be an integer > 2. Now we call the points of )Cn(F) the vertices and define two vertices K\ and K2 to be adjacent if rank(JRri — K2) = 2. Then we obtain the graph ofnxn alternate matrices over JF, denoted by T()Cn(F)). From Propositions 4.3 and 4.5 we deduce immediately Proposition 4.46: Let F be a field of any characteristic and n be an integer > 2. Then the graph T(tCn(F)) is connected, distance-transitive, and with diameter [|]. When n = 2 or 3, T(/C n (F)) is a complete graph. □ Theorem 4.17 can be interpreted as follows:

Chapter 4. Geometry of Alternate

212

Matrices

Theorem 4.47: Let F and F ' be fields of any characteristic, n and n' be integers > 4. If there is a graph isomorphism A from the graph T(ICn(F)) to the graph r(/C n /(F')), then n = nl', F is isomorphic to F ' , and when n > 5, .4 is of the form (4.20) A(X) = ptpX'P where p G F'*, P G GLn(Ff), r(/C n (F'));

when

+ K0 for all X G r(/C n (F)), a is an isomorphism from F to F ' , and if0 €

n = 4, A is of the form (4.21)

>t(X) = ^ t P(X*)
XeT()C4{F)),

where X i—► X* is either the identity map of tC4(F) or the map (4.4) of Kn{F). Conversely, any map from T(fCn{F)) to T(lCn(F')) of the form (4.20) when n > 5 or of the form (4.21) when n = 4 is a graph isomorphism. □ Corollary 4.48: Let F be a field of any characteristic and n an integer. If n > 5, then the group of graph automorphisms of the graph T(ICn(F)) consists of the following graph automorphisms Xt—tptpX'P

+ Ko for all X G £ n ( F ) ,

where p G F*, P G G L n ( F ) , cr is an automorphism of F , and i^o € r(/C n (F)). If n = 4, then the group of graph automorphisms of the graph T(/C4(F)) consists of the following graph automorphisms Xt—tptpiXyP

+ Ko for all X G Kn{F),

where X i—> X* is either the identity map of K,±(F) or the map (4.4) of

Kn{F).

□

Similarly, we call the points of VJCn(F) the vertices and define two vertices W\ and W2 to be adjacent if Wi + W2 is of dimension n + 1. Then we obtain the #rap/i 0/ £Ae protective space of n x n alternate matrices over F , which is denoted by T(V)Cn(F)) and is also called the dual polar graph of type Dn. From Propositions 4.25 and 4.26 we deduce Proposition 4.49: Let F be a field of any characteristic and n be an integer > 2. Then the graph T(VKn(F)) is connected, distance-transitive, and with diameter n. □

4.6. Application to Graph Theory

213

Theorem 4.33 can be interpreted as follows: Theorem 4.50: Let F and F1 be fields of any characteristic, n and n' be integers > 3. If there is a graph isomorphism A from the graph T(VICn(F)) to the graph T(VJCni(F')), then n = nl', F is isomorphic to F ' , and the graph isomorphism is of the form (4.31) W »—> WaT

for all W G

VKn(F),

where a is an isomorphism from F to F ' , and T G G02n(F'). Conversely, any map from T(VKn(F)) to T(VKn(F')) of the form (4.31) is a graph isomorphism. □ Corollary 4.51: Let F be a field of any characteristic and n an integer > 3. Then the group of graph automorphisms of the graph T(VJCn(F)) consists of the following graph automorphisms W i—> W°T

for all W G

VKn{F\

where a is an automorphism of F and T G G02n(F).

□

Moreover, if we call the points of VK,+(F) (or VIC~(F)) the vertices and define two vertices W\ and W2 to be adjacent if dim(W/i + W2) = n + 2, then we obtain the graph of the irreducible space V)C+(F) (or VJC~(F)), which is denoted by T(VIC+(F)) (or T(V1C-(F)), respectively). They are also called the half dual polar graph of type Dn. From Propositions 4.40 and 4.41 we deduce Proposition 4.52: Let F be a field of any characteristic and n be an integer > 2. Then the graph T{VK,+(F)) and T(V1C^(F)) are connected, distance-transitive, and with diameter [|]. When n = 2 or 3, the graph T(V1C+(F)) and T(V1C-(F)) are complete graphs. □ Theorem 4.44 can be interpreted as follows: Theorem 4.53: Let F and F' be fields of any characteristic, n and n' be integers > 5. If there is a graph isomorphism A from the graph T(VIC+(F)) to the graph T{VK+,(Ff)) (or from T(VK-(F)) to the graph T^K^F1))), then n = n7, F is isomorphic to F', and the graph isomorphism is of the

Chapter 4. Geometry of Alternate

214

Matrices

form (4.38) W i—♦ W°T

for all W £ VK${F)

(or

VK,-(F)),

where a is an isomorphism from F to F\ and T £ GO^^F'). If there is an isomorphism 4 from r(P/C+(F)) to r(P/C-,(F')) (or from r ( P / C " ( F ) ) to T(VJC^i(F'))), then n = n', F is isomorphic to F", and the graph isomor­ phism is of the form (4.39) W i—♦ W°T

for all W £ VK$(F)

(or

VK~{F)),

where a is an isomorphism from F to F ' , and T is an improper generalized or­ thogonal matrix over F ' . Conversely, any map of the form (4.38), where T £ GOtn(F'), is a graph isomorphism from T(V1C+(F)) to T{VK+(F')) (and from T(V1C-(F)) to r ( P / C " ( F / ) ) ) , and any map of the form (4.39), where T is an improper generalized orthogonal matrix, is a graph isomorphism from T(VICi(F)) to T{VK-n{F')) (and from T(V1C;(F)) to T ( ^ + ( F ' ) ) ) . □ Corollary 4.54: Let F be a field of any characteristic and n an integer > 5. Then the group of graph automorphisms of the graph T(VK+(F)) or T{VK,~(F)) consists of graph automorphisms of the following form W ■—> W*T

for all W £ T(V!C+(F)) (or

where a is an automorphism of F and T £ GO^F).

T(VIC;(F))), □

When n = 4, we can make the same remark for Theorem 4.53 and Corollary 4.54 as we made for Theorems 4.43 and 4.44. More generally, for the (2ra -f 6) x (2n + 6) matrix S defined by (4.32) and G defined by (4.33) call the elements of A^n the vertices and define two vertices W\ and W2 to be adjacent if dim(Wi + W2) = n + 1, then we obtain a graph which is denoted by T(Nn), and Theorem 4.34 can be interpreted in graph theory language as follows. Theorem 4.55: Let F be a field of any characteristic, n be an integer > 3, S be the (2n + 8) x (2n + 8) matrix defined by (4.32), and G be defined by (4.33). Then the group of graph automorphisms of the graph T(Nn) consists of graph automorphisms of the form (4.33) W*—>WaT

for all W £ T(iVn),

4.7.

Comments

215

where a is an automorphism of F and T G G02n+s(F).

n

Similarly, call the elements of iV+ (or N~) the vertices and define two vertices W\ and W2 to be adjacent if dim(W/i + W2) = n + 2, then we obtain a graph which is denoted by I\7V+) (or f (JV~)), and Theorem 4.45 can be interpreted as follows. T h e o r e m 4.56: Let F be a field of any characteristic, n be an integer > 5, S be the (2ra + 5) x (2ra + 5) matrix defined by (4.32), and G be the matrix defined by (4.33). Then the group of graph automorphisms of the graph T(N+) (or T(N~)) consists of graph automorphisms of the form (4.40) W »—► W°T

for all W G T(N+) (or W G iV"),

where a is an automorphism of F and T G G^Jn+^C^)-

4.7

□

Comments

The fundamental theorem of the geometry of skew-symmetric matrices of even order 2m over C o r E , where m > 2, was proved by L. K. Hua, but he assumes that the bijective map from /C2 m (Q (or /C2m(R)) to itself satisfies more conditions than only preserving the adjacency, cf. Hua 1945, 1946. The map (4.4) which preserves the adjacency was found by him. The funda­ mental theorem was extended to the geometry of alternate matrices of any order n > 4 over any field F by M. Liu in her thesis, cf. Liu 1966, in which only preserving the adjacency is assumed. The normal form of the maximal sets in /C n (F), i.e., Proposition 4.8 in Section 4.2 is due to M. Liu, cf. Liu 1966. The study of the intersection of maximal sets given in this section is simpler than the original one given by M. Liu in her thesis. The proof of the fundamental theorem of the geometry of alternate matrices given in Section 4.3 follows Liu 1966. Both Theorems 4.34 and 4.45 in Sections 4.4 and 4.5, respectively, are due to W. L. Chow for the case of characteristic not two and to J. Dieudonne for the case of characteristic two, cf. Chow 1949 and Dieudonne 1951. The

216

Chapter 4. Geometry of Alternate

Matrices

fundamental theorem of the projective geometry of alternate matrices over any field, i.e., Theorem 4.28, is a special case of Theorem 4.34 and M. Liu rederived it from the fundamental theorem of alternate matrices under the restriction n > 5 for the case of characteristic not two in her thesis. We followed Liu's derivation in Section 4.4 with some modification and sup­ plemented the proofs of the cases left by her. The fundamental theorem of the geometry of irreducible spaces of VtCn(F), i.e., Theorem 4.43, is a special case of Theorem 4.45. In Section 4.5 we rederived it also from the fundamental theorem of the geometry of alternate matrices. The results in Section 4.6 are merely translations of some results of the preceding sections into the graph theory language.

Chapter 5 Geometry of Symmetric Matrices 5.1

The Space of Symmetric Matrices

Let F be an arbitrary field and n be an integer > 2. Denote the set of n x n symmetric matrices over F by «Sn(F), call it the space of n x n symmetric matrices over F , and call the elements of Sn(F) the points of Sn(F). There is a group acting on Sn(F) which consists of all transformations of the form X ■—> lPXP + S

for all X G Sn(F),

(5.1)

where P G GLn(F) and S G Sn(F). Denote this group by GSn(F) and call it the group of motions of Sn(F). Our problem is to characterize the elements of the group of motions GSn(F) by as few geometric invariants as possible. Parallel to Propositions 3.1 - 3.3 and Definition 3.1 of Chapter 3, we have Proposition 5.1: GSn(F)

acts transitively on Sn(F).

□

Definition 5.1: The arithmetic distance between two points S\ and S2 of Sn(F),

denoted by ad(5i, 52), is defined to be the value of rank(Si — 52).

If ad(Si, £2) = lj then they are said to be adjacent Proposition 5.2: Let Su S 2 , S3eSn(F).

Then

1° ad(5i, S2) > 0; ad(Si, S2) = 0 if and only if Si = S2. 217

□

Chapter 5. Geometry of Symmetric

218

2° ad(5i, 5 2 ) = ad(5 2 , ft). 3° ad(5i, S2) + ad(S 2 , S3) > ad(5i, S3).

Matrices

□

Proposition 5.3: The elements of the group GSn(F) leave the arithmetic distance between any pair of points of Sn(F) invariant. □ However, for any r with 1 < r < n, the set of pairs of n x n symmetric matrices over F for which the arithmetic distance between any pair is r does not necessarily form an orbit of GSn(F). This is because two n x n symmetric matrices of the same rank are not necessarily cogredient. It follows from Proposition 5.3 that the arithmetic distance between any pair of points of Sn(F) is a geometric invariant under the group GSn(F), so is, in particular, the adjacency. We will see that the adjacency alone is sufficient to characterize the transformations of the form (5.1) to within automorphisms of F and scalar multiplications by elements of F*. More precisely, we have the fundamental theorem of the geometry of symmetric matrices over any field. Theorem 5.4: Let F be a field of any characteristic and n be an integer > 2. Then any bijective map A from Sn(F) to itself, for which both the map A and its inverse A'1 preserve the adjacency in Sn(F) is of the form = a%PXaP + So

A(X)

for all X G Sn(F),

(5.2)

where a G F*, P G GLn(F), a is an automorphism of F , and S0 G 5n(F), unless n = 3 and F = F 2 . When n = 3 and F = F 2 , there is an extra bijective map of the form £ll

^12

#13

#11

£12

^13 \

#12

^22

0

^13

0

^33

£12 ^13

£22 0

0 X33)

Xu

^12

^13

X\l + 1

^12 + 1

#13 + 1

^12

#22

#22

1

1

1 x33

£12 + 1

^13

x13 + 1

1

^33

(5.3)

for all £ u , Zi 2 , x13, x 22 , x33 G F 2 , from <S3(F2) to itself and A is a product of bijective maps of the form (5.2) and this extra bijective map (5.3). Con­ versely, any map of the form (5.2) or (5.3) from Sn(F) to itself is bijective and both the map and its inverse preserve the adjacency. □

5.1.

The Space of Symmetric Matrices

219

The proof of Theorem 5.4 will be given in Section 5.4 for the case when F is of characteristic not two, in Section 5.6 for the case when F is of characteristic two and F ^ F 2 , and in Section 5.7 for the case when F = F 2 , respectively. Similar to Definition 3.2, we have Definition 5.2: Let S, S' G Sn(F). When S ^ S', the distance d(S, S') between S and S' is defined to be the least positive integer r for which there is a sequence of r + 1 points So, Si, S 2 , • • •, Sr with So = S and Sr = S' such that Si and S;+i are adjacent for i = 0, 1, 2, • • •, r — 1. When S = S', define d(S, S) = 0. □ In contrast to Proposition 3.5, we have Proposition 5.5: For any two points S and S' G <Sn(F), ad(S, S') < d(S, S'). When F is of characteristic not two, the equality sign holds, i.e., ad(S, S") = d(S, S'). When F is of characteristic two, ad(S,

f d(S, S') if 5 - 5 ' is nonalternate, S')=l [ d(S, S') - 1 if S - S' is alternate.

Proof: Suppose that d(S, SO = r. Then there is a sequence of points of Sn{F) So = S, Si, S2, ''' f S r = b such that Sz and S;+i are adjacent for i = 0, 1, 2, • • •, r — 1. That is, rank(St- — S t +i) = 1 for i = 0, 1, 2, • • •, r — 1. By Proposition 1.21, r-l

ad(S, SO - rank(S - SO < £ rank(St- - Sw)

= r = d(S, S').

t=0

The first assertion is proved. Now assume that F is of characteristic not two. Suppose that ad(S, SO = r ; , i.e., rank(S - SO = r'. By Corollary 1.33 there is a P £ GLn(F) such that

220

Chapter 5. Geometry of Symmetric

Matrices

*P(5 — Sf)P is a diagonal matrix *P(S - 5')P S')P = [a , o ] 0], , *P(5 [ai, r/, 0,0,• •••••, u a 2 , • ••,•,a
(5.4)

where ai, a2, • • •, a r ' € F*. Let 1 2, 2, •• •■ ,•r ' .•, r'. ft ^ _ _1 1))K, K , «2, • ••,•,a;, a,-,0,0,••• ••. O •, J0P]- P," \t =i =l, 1, Ri== tP a2, ••

Then «So

=

*~>5 6 i = 6 — . i t i ,

«->2 — *-> — ^ 2 ?

* ■• ,

Srt = S — Rri

is a sequence of points of Sn(F) such that Si and S^i are adjacent for i = 0, 1 , 2 , - - •, r' - 1. It follows that d(S, S") < r' = ad(5, S'). Combining with the first assertion, we obtain ad(5, 5') = d(S', S'). Finally, assume that F is of characteristic two. We distinguish the following two cases. Consider first the case when S—Sf is nonalternate. Assume that ad(S—Sf) = r', i.e., rank(5 - S") = r'. By Proposition 1.35, there is a P <E GLn(F) such that *P(5 — S')P is a diagonal matrix of the form (5.4). As in the characteristic not two case, we can show that ad(5, S') = d(5, S"). Then consider the case when S — S' is alternate. By Proposition 1.34 rank(5 — Sf) is even, let it be 23, and there is a P G GLn(F) such that

(°

1/ 0 t S5 - S' = *P P\ /W J(s)

J/(*) « 0

>\I

I"'

P.

Q(n-2s) i

Thus S — Sf + tPEnP is a nonalternate matrix of rank 2s. By what we have proved in the preceding case,

= ad(S, ad(S, S' S' --- 'PE^P) S' --• t'PBnP) PE11P) = d(5, 5' *PEuP) = = 2s. 2s. Thus t d(S, S') < d(5, d(S, 5" S' --■tPE PE d(S'- ■tPEnP, *PEnP,S')S')==2s2s++l.1. 11P) nP) ++d(S'

By the first assertion of this proposition

ad(S, S') < d(5, ad(5, S') d(5, 5"). S').

5.1.

The Space of Symmetric

Matrices

221

But ad(S, S") = 2s. Therefore d(5, Sf) = 2s or 23 + 1. We assert that d(5, S') = 2s is impossible. Suppose that d(5, Sf) = 25, then there is a point W € <Sn(F) such that d(5, W) = 1 and d(W, 5') = 25 - 1. Let Q = S — W, then rankQ = 1. By the first assertion of the proposition ad(W, 5') < d(W, S") = 2 5 - 1 . On the other hand, ad(W, 5') = rankfW - 5') = rank(5 - Sf - Q). Therefore rank(5 - 5" - Q)

<2s-l.

Let Q1 = ' ( P - ^ Q P - 1 , then rankQi = 1 and rank( t (P" 1 )(S' - S^P'1 Qi) < 2s — 1. It can be easily computed that the 25 x 25 minor situated at the upper left corner of

( 0 (P-1)(S-S')P-1-Q1

I(s)

= I

/« 0

\-Q1 g(n-25)

is 1, which contradicts r a n k ( t ( P " 1 ) ( 5 - S")^" 1 ~ Qi) < 25 - 1. Therefore d(5, 5') = 25 + 1. D

This proposition shows that for the geometry of symmetric matrices there are some differences between the characteristic not two case and the charac­ teristic two case. In the following we shall treat these two cases separately. From Proposition 5.5 we deduce Corollary 5.6: Let F be a field of characteristic not two and A be a bijective map from Sn(F) to itself. Assume that both A and A'1 preserve the adjacency in Sn(F), then A also preserves the arithmetic distance in Sn(F). □

222

5.2

Chapter 5. Geometry of Symmetric

Matrices

Maximal Sets of Rank 1

In this section let F be a field of any characteristic and n > 2. Recall that Eij denotes the n x n matrix whose (z,j)-entry is 1 and all other entries are O's. Definition 5.3: A subset Ai of Sn(F) is called a maximal set of rank 1, if any pair of points of Ai are adjacent and if there is no other point of Sn(F) outside Ai, which is adjacent to each point of Ai. □ Proposition 5.7: A maximal set of rank 1 is transformed into a maximal set of rank 1 under a transformation of the form (5.1). Q Let Mi = {xEu\xeF},

» = l,2,.--,n.

(5.5)

Proposition 5.8: All Af t 's for i = 1, 2, • • •, n are maximal sets of rank 1. Moreover, every maximal set of rank 1 can be transformed under a transformation of the form (5.1) to Ai\. Proof: We prove only the second statement, the proof of which gives also a proof that M\ is a maximal set of rank 1. Let AI be a maximal set of rank 1. Assume that So G Ai. Then the transformation X ■—► X - So,

(5.6)

which is of the form (5.1), will carry M to a maximal set of rank 1 containing the zero matrix 0. Denote the latter by At'. Let Si G M.' and S\ ^ 0. Then S\ is of rank 1. There exists a P G GLn(F) such that *P5iP = [a1, 0, 0, . . - , 0], where ax G F*. After performing the transformation X i-> tpXP, which is also of the form (5.1) and leaves 0 fixed, Ai' is carried into a maximal set of rank 1 containing 0 and aijEn. Denote the latter by AI". Let S G Ai", S + 0, and S ± mEn. Write S — (-Sij)l
5.2. Maximal Sets of Rank 1

223

Since 5 is adjacent to both 0 and aiEu, rank 5 = rank(5 — aiEu) = 1. It follows that Sij = 0 for 2 < i < j < n. Since rank 5 = 1, S\j = 0 for 2 < j < n. Therefore 5 = 5 n £ n . Hence M" = Mi. □ Corollary 5.9: Let Su 5 2 G Sn(F) be such that ad(5i, 5 2 ) = 1. Then there is a unique maximal set of rank 1 containing both Si and 52, which is the set of matrices {xSi + (l-x)S2\xeF}.

□ Corollary 5.10: If there are two distinct maximal sets of rank 1 whose intersection is nonempty, then their intersection contains a single point. □ Corollary 5.11: Any maximal set of rank 1 can be put into the following general form { x *(ai, a 2 , • * *, On)(«i, «2, • * *, cLn) + So I x e F }, where (ai, a 2 , • • •, a n ) is an n-dimensional nonzero row vector over F and So is an element of Sn(F). □ Proposition 5.12: Let M and M' be two distinct maximal sets of rank 1 and assume that Mf\M' ^ (j). Then they can be transformed simultaneously under a transformation of the form (5.1) to Mi and M2Proof: By Proposition 5.8 we can assume that M = Mi. By Corollary 5.10, Mi fl M' consists of a single point, let it be called So. Then the transformation (5.6) leaves Mi invariant and carries So into 0. Thus we can assume that Mi fl M' — { 0 }. Let 5 G M', 5 ^ 0 , and write V %u

S22 J

where sn G JP, U is an (n — l)-dimensional row vector over F , and 522 is an (n — 1) x (n — 1) symmetric matrix. Since ad(5, 0) = 1, rank 5 = 1. If 522 = 0, then we must have u = 0 and 5 G Afi, we get a contradiction. Therefore 522 i1 0 and is of rank 1. There is a Pi G GLn-i(F) such that t

Pi5 2 2 Pi = K

0, 0, . . . , 0],

Chapter 5. Geometry of Symmetric

224

Matrices

where a 2 G F*. After performing the transformation

*~r P>C *)• which is of the form (5.1) and leaves Ai\ fixed, we can assume that M! contains 0 and / a n a i 2 «i3 ••• « i n \ Ci2

C2

0

• • •

0

T = Uis

0

0

•••

0

\flm 0 0 ••• 0 / Since rank T = 1, we must have a 13 = • • • = a i n = 0. Thus an T = I a12

ai2 a2

Q (n-2)

where anfl2 - a*12 ?2 = 0- Under the transformation */ X

1 -ai 2 aj

1 X I —a^aj1

1 7(n-2)

1 j(n-2)

which is of the form (5.1) and leaves J M I fixed, we can assume that M! contains 0 and a2i?22? where a2 G F*. By Corollary 5.9, Ai' = M.2-

5.3

n

Maximal Sets of Rank 2 (Characteristic Not Two)

In this and the next section we assume that JF is a field of characteristic not two. Definition 5.4: Let F be a field of characteristic not two. A subset C of Sn(F) is called a maximal set of rank 2, if the following conditions are satisfied: 1° C contains a maximal set of rank 1, denoted by M. 2° For any S G C\M and T e M, ad(5,T) = 2. 3° For any S G 5 n ( F ) , ad(5, T) = 2 for all T G M implies 5 G £ . □

5.3. Maximal Sets of Rank 2 (Characteristic Not Two)

225

Proposition 5.13: A maximal set of rank 2 is transformed into a maximal set of rank 2 under a transformation of the form (5.1). □ Let f/

0 0

Ci =

•••

0

xu Xu

0

Xi-l,i

0

•••

0 \ 0

0

*£lt)

•£«—1,1

0

* * 5 #« —1,1) ^ « )

•E«,t+lr

0

? ^*n G -*

o /

Vo

1 < i < n.

(5.7)

Then we have Proposition 5.14: All £ z 's, where i = 1, 2, •••, n, are maximal sets of rank 2. Moreover, any maximal set of rank 2 can be transformed under a transformation of the form (5.1) to the set C\. Proof: We prove only the second statement, the proof of which gives also a proof that C\ is a maximal set of rank 2. Since each d can be transformed into C\ under a transformation of the form

*PXP,

X

where P is a permutation matrix which permutes the first and z-th row (and column), by Proposition 5.13, £, is also a maximal set of rank 2. Let £ be a maximal set of rank 2 and M be a maximal set of rank 1 contained in C. By Proposition 5.8 we can assume that M = Mi. Let S G C\M\, and write

s = (*>)i<«.,-<» = (*"

gj-

Since ad(5, 0) = 2, S is of rank 2. Thus rank5 2 2 < 2. If rank £22 — 2, then there is a Pa G GLn-\ (F) such that t

PlS22Pl=

[

2

Q(„-3)J,

Chapter 5. Geometry of Symmetric

226

Matrices

where T2 is a 2 x 2 nonsingular symmetric matrix. After performing the transformation Jfh- ' ( ' « ) * ( ' * ) ■

which leaves Mi fixed, we can assume that C contains

T =

/ su

L

(T2

{

v

\

\

Q(»-3) J j

>

where v = uP\. Since ad(T, 0) = 2, T is of the form ( sn

T =

Sl2

S12

\

«13

T2

Sl3

Q(n-3) j

Since ad(T, XEu) = 2 for every A € F, we must have IT2I = 0, this is a contradiction. If rank 522 = 1, then there is a P2 € GLn-i(F) such that

%S %S2222PP22 = [
X*-

1

l

A M *)•

which leaves .Mi fixed, we can assume that C contains

\

( Su

/a2

T = ( ,

o j / '

where v = uP2. Since ad(T, 0) = 2, T is of the form

T =

Mu

512

5i2

«2

\ Q(n-2) ) '

Since a2 ^ 0, we can find an element \0 € F such that ad(T, X0En) = 1, this is also a contradiction. Therefore rank£22 = 0, i.e., 522 = 0. Hence 5 € A . Therefore £ C £ 1 B

5.3. Maximal Sets of Rank 2 (Characteristic Not Two)

227

Clearly for any Si G A \ A < i we have ad(5i, T) = 2 for all T e M = Since C is a maximal set of rank 2, £ i C C. Hence C = C\.

Mx. □

C o r o l l a r y 5.15: Any maximal set of rank 1 is contained in a unique max­ imal set of rank 2. The unique maximal set of rank 2 containing M\ is £i. □ C o r o l l a r y 5.16: Any maximal set of rank 2 can be put into the following general form /£ii

a?i2

&12

Xln\

0

0

\xln where P G GLn(F)

P + So

^llJ^12>-"5a?ln €

P

(5.8)

0 0 / and So is an element of

Sn(F).

D

The converse of Corollary 5.15 is not true. In fact, a maximal set of rank 2 contains several maximal sets of rank 1. However, we have the following proposition. Proposition 5.17: Let £ be a maximal set of rank 2 and 5 0 G £ , then there is one and only one maximal set of rank 1 contained in C and containing So. Proof: By Proposition 5.14 it is sufficient to consider the case C = C\. Let M. be any maximal set of rank 1 contained in C\ and containing So. If So = 0, then every element of J M \ { 0 } is of rank 1. Therefore M = M\. If 5o 7^ 0, then the transformation (5.6) leaves C\ invariant, carries So to 0, and M to M — So. By the above case, M — So = Mi. Therefore M=M1 + S0. a Proposition 5.18: Any two distinct maximal sets of rank 2 which have a nonempty intersection can be carried simultaneously by a transformation of the form (5.1) to C\ and £2Proof: Let C and C be two distinct maximal sets of rank 2 which have a nonempty intersection. Let So G C fl C. By Proposition 5.17 there is a unique maximal set of rank 1, denoted by M, which is contained in C and contains S^, and there is also a unique maximal set of rank 1, denoted by Mt\ which is contained in C and contains SQ. Since C ^ £', by Corollary

228

Chapter 5. Geometry of Symmetric

Matrices

5.15 M 7^ M'. By Proposition 5.12, M and M! can be carried simultane­ ously by a transformation of the form (5.1) to M.\ and M.2, respectively. Then by Corollary 5.15, C and C are carried simultaneously to C\ and £2, respectively. □ Proposition 5.19: Let F b e a field of characteristic not two and 1 < i < j < n. There exist exactly two maximal sets of rank 2 which contain 0 and Eij + Eji in common. More precisely, they are d and Cj. Proof: It is enough to consider the case (z, j) = (1, 2). The other cases can be treated in the same way. It is clear that C\ and C2 are two maximal sets of rank 2 which contain 0 and E\2 + E2i in common. Now let C be any maximal set of rank 2 which contains 0 and E\2 + E2\. By Proposition 5.17, let M be the unique maximal set of rank 1 contained in C and containing 0. Then every element distinct from 0 in M is of rank 1. Let 5 G M, 5 ^ 0, and write S in the block form S = ( ^u ^12 ^ V S12 S22 J ' where S\\ is a 2 x 2 symmetric matrix, S12 is a 2 x (n — 2) matrix, and S22 is an (n — 2) x (n — 2) symmetric matrix. Since S is of rank 1, ^ n is of rank < 1. We distinguish the following cases: (a) S\\ = 0. Since rank S = 1, we have 5i2 = 0. Then rank 522 = 1. Thus S — (Ei2 + E2i) is of rank 3, i.e., ad(5, Ei2 + E2\) = 3, which contradicts the assumption that £ is a maximal set of rank 2. Therefore the case Su = 0 cannot occur. (b) Sn =fi 0. Since 5ii is a 2 x 2 symmetric matrix of rank 1, there are only three possibilities:

* - ( • 0). (° .)• (; A 0 In the first two cases s ^ 0 and in the third case st ^ 0. For the first case from rank 5 = 1 and ad(5, E12 + £21) = 2 we deduce 5 = -s^n, then by Corollary 5.9 M = Mi and by Corollary 5.15 C = C\. For the second case we have 5 = sE22, then M = M2 and £ = £2. These two cases are what we

5.3. Maximal Sets of Rank 2 (Characteristic Not Two)

229

want. It remains to prove that the third case cannot occur. Suppose that

s-H2)'

*>-(;

where st ^ 0. By Corollary 5.9 we can assume that 3 = 1. From rank 5 = 1 and ad(5, E\2 + E21) = 2 we deduce

(I

t

\

s = [t e . V oj Let P = / + tEi2, then the transformation 1 x ■—-+ ► ' ( p'{p-^xip- 1 ) * ^ - 1)

X H -

carries S into En. By Corollary 5.9 there is a unique maximal set of rank 1 containing 0 and .En, which is A4i, and by Corollary 5.15 there is a unique maximal set of rank 2 containing A4i, which is C\. It follows that M = lPM\P and C = lPC\P. Then C consists of elements of the form /

Zll

tXn + £12 #13

V \

Xin sin

* ■ •

xln

t2xn + 2txu tXi3 tx13 0

•''

tXin

••

0

0

.. •••

tXn + £12

^13

tXi txlnn

\

)

0 )/

where a?u, £12, • • •, Xin G F. Clearly, Ei2 + E2i is not an element of the above form. Hence E12 + E2\ £ C. We obtain a contradiction. □ Now let us study the intersection of two maximal sets of rank 2. Let C and C be two distinct maximal sets of rank 2 and assume that C C\ C ^ . By Proposition 5.14 we can assume that C = C\. Let So G C\ 0 C. Then the transformation (5.6) leaves C\ invariant. Denote the image of £ under (5.6) by C". Then 0 G C\ f) C". By Proposition 5.17 there is a unique maximal set of rank 1 contained in C" and containing 0, which will be denoted by M". By the proof of Proposition 5.12 we can assume that t

M" =

I^vJ^r

M

-\}

■ J c «)"■

Chapter 5. Geometry of Symmetric

230 where Pi € GLn-i{F)

C" =

Matrices

and A e F. By Corollary 5.15

£

A 1 /( n-2)

(

Pi)

,-1

1

'

| A

»^iCJ

Denote the first row of Px * by (p 2 , * * * ,Pn). Then (p 2 , * * * ,Pn) 7^ 0 and f ^ -2xi2A Z12P2

d DC" = {

L\

X12P2 0

^12Pn \

0

0

X12pn

0

xueF ;

Therefore we have Proposition 5.20: Let F be a field of characteristic not two, £ ' be a maximal set of rank 2 and assume that C\ ^ £ ' and C\ 0 £ ' ^ . Then £1 fl £ ' is of the form r

dr)Cf= I

X

(pi Pi

P2

Pn\

0

0

\Pn

0

oj

+ 50 i e F

where pi,p 2 , • * * ,Pn £ F, (p 2 , * * * ,Pn) 7^ (0, • • •, 0), and So is an element of £1 0 £'. D Proposition 5.20 suggests the following definition. Definition 5.5: Let F be a field of characteristic not two and £ be a maximal set of rank 2. A line in £ is either a maximal set of rank 1 contained in £ or the intersection of £ and another maximal set of rank 2 when it is nonempty. D From Proposition 5.20 and Definition 5.5 we deduce Proposition 5.21: Let F be a field of characteristic not two and £ be a maximal set of rank 2, then £ has an n-dimensional affine space structure over F with the lines defined in Definition 5.5. More precisely, let £ be (5.8),

5.4. Proof of the Fundamental Theorem (I)

231

then the map C / Xu

tp

X12

\xln

—> AG(n,

X12

'••

0

•••

0

0

•••

0 /

F)

Xin\

P + So

I

►

{xiuXi2,'-',Xin)

is a bijective map which carries lines in C into lines in the n-dimensional arrine space AG(n, F). □

5.4

Proof of the Fundamental Theorem (I)

After the foregoing preparations now we come to the proof of the fundamen­ tal theorem of the geometry of symmetric matrices for the case of charac­ teristic not two. Proof of Theorem 5.4 (the Case F Being of Characteristic Not Two): Let F b e a field of characteristic not two and n be an integer > 2. The second statement in Theorem 5.4 is obvious. We prove only the first statement. We proceed in steps. (i) Let A : Sn(F) —> Sn(F) be a bijective map and assume that both A and A'1 preserve the adjacency. By Corollary 5.6 A preserves also the arithmetic distance. Clearly, the bijective map Ax : Sn{F)

—>

Sn(F)

X ^

A(X)-A{0),

where 0 = 0^n\ preserves also the arithmetic distance and carries 0 to 0. If we can prove that Ai is of the form (5.2), then so is A. Therefore we may assume that .4(0) = 0.

(5.9)

(ii) Mi is a maximal set of rank 1 containing 0. Since A preserves the arithmetic distance, by Definition 5.3 and (5.9) A(A4i) is also a maximal

Chapter 5. Geometry of Symmetric

232

Matrices

set of rank 1 containing 0. By Proposition 5.8, after subjecting A to a transformation of the form (5.1), besides (5.9) we can assume that Mi. A(Mi) A(Mi) === Mi. By Corollary 5.15, C\ is the unique maximal set of rank 2 containing Therefore we also have

M\.

A(d) -== &. Aid) A. Clearly A induces a bijective map from C\ to itself which carries lines into lines. By Proposition 5.21 C\ has an affine space structure of dimension n over F. Let

/xn

A

X\2

\xln

0

•

0

/(yn 2/n 2/12 2/12 • • yin\ yin) 2/12 0 • • 0

0

• ■■

0 I

\yin \2/ln

' "

X\2

Xln\

0

•

• o)

Then by the fundamental theorem of the affine geometry (i.e., Theorem 2.7), we have a

= =( z(X i l 1U , Z lX12, 2 , " • ,*« m (S/ll, 2/12, * *,2/ln) ' * , 2/ln) • ,) Xf flnP) (2/11, 2/12,"

P + (bU•■6•2, ,6 ' ' n*),, 6 n ), +(61,62,

where a is an automorphism of F , P G GLn(F), and &i, 62, • * *, K G F. Since A(0) = 0, we have (6i, 62, • • •, bn) = 0. Since A(Mi) = Mi, P is of the form

p=

p„-. , ) •

where pu £ F*, u is an (n — l)-dimensional row vector over F and P n _i 6 GLn-i(F). After subjecting A to the bijective map Y>—■* [Pn '(-

Pn

M

Pn-

(-

^-1

Pn 2^11

r

n-l

*U

Pn-,

) ) ' " ■

we can assume that

^ ( X ) == X for all * € A .

(5.10)

(iii) It follows from (5.10) that A(EU + Ea) = Eu + E{1 for every i with 1 < i < n. By Proposition 5.19 we have A(£i) = £i, t = 2,3,- •,n,

(5.11)

5.4. Proof of the Fundamental Theorem (I)

233

where d is denned by (5.7). By (5.9) and Proposition 5.17 A(Mi)

= Mi, i = 2,3, • • •, n.

(5.12)

We assert that A(X)

= X

for all X G £ t , * = 2,3, • • •, n.

(5.13)

Consider first the case n = 2. By (5.11) we can assume that

*12W°, I 2 ).

A(°

\x12 x22J \xl2 x\2) As in Step (ii), A induces a bijective map from £2 to itself which carries lines into lines. By Proposition 5.21 and the fundamental theorem of the affine geometry (zi2> Z22) = (^12, X22)°2P

+ (612, 622),

where <72 is an automorphism of F , P G GL2(F), and 612, 622 £ F. Since ,4(0) = 0, we have (b12, b22) = 0. From (5.11) we deduce 4 ( £ x D £ 2 ) = £1 H £2. By (5.12) we have also 4 ( ^ 2 ) = A^2- It follows that P is a diagonal matrix. Let P = [p1? p 2 ] ? where p 1? p 2 £ P*- Then X

12 — ^12^1?

X

22 =

X

22^2

for

all X 1 2 , #22 G P

Since £i 2 (Pi2 + #21) € £1 and A(X) = X for all X G £ 1 , we have Zi 2 = x ilPi- L e t £12 = 1, we obtain pi = 1. Hence 0*2 = 1. Since X22F22 and —x 2 2(Pn +^12 + ^21) are adjacent, 4.(^22^22) = ^22P2£"22 and 4(—£22(^11+ P12 + P21)) = — x22(Fn + E12 + E2i) are adjacent too. It follows that p2 = 1. This proves that X l 2

A(° \#12

^22/

*12)

W ° V^12

for all * „ , x 22 € F.

#22/

Then consider the case n > 3. For simplicity of writing, we consider only the case n = 3, since the general case can be treated in a similar way. By (5.11) we can assume that / 0 A x12 \ 0

x12 x22 x23

0 \ / 0 £23 = x\2 0 / \ 0

x*12 0 \ x22 x23 x*23 0 /

234

Chapter

5.

Geometry

of Symmetric

Matrices

and 0 0 ^13

0 0 #23

sis \ x23 \= ^33/

/ 0 0 V^IS

0 0 X 2Z

x'13 x**3 X 33 ,

for all #i2, £22, x 23, #13, ^33 £ ^ - As in (ii) A induces a bijective m a p from £2 t o itself and one from £ 3 to itself, which carry lines into lines. By Proposition 5.21 and t h e f u n d a m e n t a l t h e o r e m of t h e afrme geometry we have (^125 #22> X23) = (X12> ^22, £23)^^2 + (&12, 622, &23) and (Zl3> X23> X33) = ( X 13, ^23, X33Y3P3 + (C13, C23, C33), where 02 a n d
and 612,622, fc23,ci3, 235C33 € F .

GL3(F),

By ,4(0) = O w e have (&i2, 622, 623) =

(cis, c 2 3 , c 33 ) = 0. From (5.10) and (5.11) we deduce A(& for i ^ j , and we have (5.12) A(Mi)

f) Cj) = & C\ Cj

= Mi (t = 2, 3). It follows t h a t b o t h P 2

and P3 are diagonal matrices. Let P 2 = [P21, P22, P23] and P3 = [p 3 i, p 3 2 , P33], where p 2 j , P3j £ F* (j = 1, 2, 3). T h e n ^12

=

X

12P21,

^22 -

^22^22,

#23 = ^23^23

for

all X i 2 , £ 2 2 , #23 €

F

^23 = ^23^32,

Z33 = ^33^33

for

all X13, X23, X33 £ F .

and X

13 = ^?|P31,

As in t h e case n = 2 we can prove t h a t p 2 i = P22 = P31 = P33 = 1 and cr2 = cr3 = 1. Clearly,

0

2

-X

0 0

and

—X

0 0

are of a r i t h m e t i c distance 2, so are r

o

U £ ZP23

#P23 I 0

and

4* —X

~2X

0 0

— X

0 0

5.4, Proof of the Fundamental Theorem (I)

235

Therefore we must have p2s = 1. In a similar way we can prove that p 3 2 = 1. Hence we have A(X) = X for all X £ d i = 2, 3. Therefore we can assume that A satisfies (5.10) and (5.13). (iv) We prove by induction on r that AyX^Ei^

+ Xi2Ei2i2 + - • + XirEirir) = A t l £ t l t l + At-2i?t-2t-2 + • • • + Atri?»rtr (1 < i i
(5.14)

For r = 1, by (5.10) and (5.13) we have, in particular, A(XiEu) = XiEu for i = 1,2, • • • ,n. Assume now that (5.14) is true for r — 1, where 2 < r < n, we are go­ ing to prove that it is also true for r. If some of A tl , At-2, • • •, A,r are zero, then (5.14) holds by induction hypothesis. Now we consider the case that all A tl , At-2, • • •, Atr are nonzero. We consider only the case ii = 1, i2 = 2, • • •, i r = r as an example. The other cases can be treated in a similar way. Assume that

J2 **iEih

A(X1E11 + X2E22 + • • • + XrErr) =

l
where x*j = x^ (1 < i, j < n). It is clear that XiEu + X2E22 -\ \- XrErr is adjacent with both XiEu + X2E22 H h A r _ijE r _i, r _i and A2i?22 + A3i?33 + \-XrErr. By induction hypothesis *4(Aii?ii + A2i?22H hA r _ii? r _i )r _i) = Ai£n + A 2 £ 22 + • • • + A r _i£ r _i, r _i and *4(A2£22 + A 3 £ 33 + • ■ • + XrErT) = A2£22 + A 3 £ 3 3 + \-XrErr. So £i<»\ j
£ l
where x*j == x^ (1 < z, j < r). We distinguish the cases r = 2 and r > 2.

x^Eiji

Chapter 5. Geometry of Symmetric

236

Matrices

(a) r = 2. We have

1fXi

A2

A

\

\

0(»-a) J

K

1 (I — 1 ^12

X

\

12

^22

Q(n-2)

j

By (5.10) and (5.13), x^ ^ 0 and x22 ^ 0. To simplify our writing we write down only the 2 x 2 submatrices situated at the upper left corner of the nxn symmetric matrices under consideration when all the other entries are 0. Thus ^(

A l

1= ( A2 ^2/

\

X l 2

Xl1 x

)

x

V^i2 \ 12

#2222/'

Since A i ^ n + A2i?22 is adjacent with both XiEu and A2^225 we have ^ n - Ai X

12

X

xu

12

X

\2

X

X

^22

= 0.

(5.15)

22 ~ A2

\2

Hence Al#22 ~~

(5.16)

^2X\\'

Since A i ^ n + A2i?22 is adjacent with both \\E\i A2^22 — Ai(i?i2 + E21 + E22), we have x

i i - Ai + A2

x\2 + A2

xn x\2 + Ai

X

«12 ^2 #12 + A2

^22 22

— A2(i?ii + £12 + ^21) and

#12 + Ai ^22 + Ai — A2

= 0.

(5.17)

From (5.15) and (5.17) we deduce —22r* Al ^IJ — Ai == ^11 x12,

A2 x22 ~ zx 1 2 A2 =— ^22

(5.18)

Substituting (5.18) into (5.16), we get 2x*

^22

= 2r* x 1 2 .

(5.19)

If Ai ^ A2, then from (5.16) we deduce x ^ ^ x*22, and from (5.19) we deduce x\2 = 0. From the adjacency of \\E\\ + \2E22 with XiEn and X2E22, we deduce x^ = Ai, x22 = A2. Hence ^(

A l

AJ =

■(*■

A2)

for Ai ^ A2.

237

5.4. Proof of the Fundamental Theorem (I)

There remains to consider the case Ai = A2. Let A = Ai = A2, then A 7^ 0. Since A(JEH + E22) and AJEH + ^£22 are adjacent for any \i ^ A, we have X

xX

12

x

22 ^22

12

= 0. rft I

Let fj, = 0 and 2A, we obtain two identities, from which we deduce x^ = A. Similarly, x\2 = A. From (5.20) we deduce x\2 = 0. Therefore - (

A

* ) ■

= (A

*)•

The case r = 2 is completely settled. (b) r > 2. We use again the fact that \\En + A2 J322 H

V A r E r r is adjacent

with both Ai£n + A2£22 + hA r _i25 r _i tr _i and A 2 £ 2 2 + A 3J E33 + VKErr, from which we deduce x*{ = A,-, 1 < i < r and x*- = 0, 1 < i < j < r. Therefore (5.14) holds also for Case (b). Thus we may assume that A satisfies (5.10), (5.13), and (5.14). (v) We prove that for any r (1 < r < n) and 1 < z'i < i2 < • • • < ir < rc, S t . t t - ^ t . .-,) A( *hAs t't? it)=== E -*( EE *i.i.Ei. E

ll<s,t
(5-21)

l<s,t
where Xisit = x ttta and x*ait — x*tia (1 < s, t < r), for all x t s t t G F (1 < s < t < r). We take the case i\ = 1, i 2 = 2, • • •, i r = r as an example. That is, we prove that x x*-E-ij-&ij) — A E x*En)= E *?, •£«> (5-22)

.4 E

l
E

l
where X{j = Xji and x*- = x^ (1 < i, j < r), for all xtJ- G F (1 < i < j < r). By (5.10), (5.22) holds for r = 1. (5.22) holds trivially also for r = n. Now consider the case 1 < r < n. Let

**(ll
=l<*\j<»i E

X

ij^Ji3'>

l
where a;^- = x^ (1 < z, jf < r) and x^ = #!-t- (1 < i, j < n). For any Ai, A2, • • •, Ar G F, Si
E *«** -EA,£« x-jEij-j:^ E i=l

ll
1=1

Chapter 5. Geometry of Symmetric

238

Matrices

is < r, from which we deduce x*st = 0 for r < s,t < n and x*st = 0 for l<s
A( Yl

£

i >itEi, w) =

l
(5.23)

X

iaif&is if)

l<s,t
where #,at-t = # tt , s (1 < s, t < r), for all Xiait E F (1 < 5, t < r) by induction on r. Let r = n, then we obtain Theorem 5.4 for the case of characteristic not two. By Step (iv), (5.23) holds for r = 1. Now let us consider the case r = 2. We take the case z'i = 1, i2 = 2 as an example. That is, we are going to prove that 'zii

£12^ xi 2y

M <xu

fxn ( #n

#x112 2

(5.24)

for all # n , X12, #22 G F. Here we use the convention made in (v) for writing matrices whose entries are 0 except possibly the entries in the upper left 2 x 2 submatrices of them. By (5.10), (5.13), and (5.14) if one of x n , #i 2 , and £22 is zero, then (5.24) holds. Let us consider the image of

f#ll f#n V#12 \£l2

# \

#12 \ i2 ##2222//

(5.25)

under ^4, where i n ^ 0, x12 ^ 0, and x22 ^ 0. By (5.21), we have ^4^11 \Zl2

\Xi2

Xl

= (x*n

A

^22/

X22J

\#t2

V^i2

^X1*A 2^ ^22 / ^22/

By (5.10), (5.13), and (5.14), # ^ ^ 0, *i2 ^ 0, and #*2 ^ 0. Since the arithmetic distance between (5.25) and XuEu + xE22 for every x G F is 2, X X

12

12

%22

*^

7^ 0 for every # G F.

It follows that #1! = x n . Similarly, we have x\\ = x22. ^ n ^ n + Xi2(E12 + E2i) are adjacent,

0

^ 1 2 ~~" ^ 1 2

^ 1 2 ~~ ^ 1 2

#22

= o,

Since (5.25) and

5.4. Proof of the Fundamental Theorem (I)

239

from which we deduce x\2 = #12- Therefore (5.24) is proved. We remark that when n = 2 Theorem 5.4 for the case of characteristic not two is completely proved. Now assume that (5.23) is true for r — 1 (3 < r < n), we are going to prove that it is also true for r. Take the case z'i = 1, i^ = 2, • • •, ir = r as an example. That is, we want to prove

E ij-^ij) ~- E E %3^*&, i3i .4l<»\j
x

x

(5.26)

J

where X{2- — Xji (1 < z, j < r), for all Xij G F (1 < i < j < r). (vii) Consider the image of

E

XijU/ij

+X

(5.27)

rrHfrr^

1< l < *i,i
where x tJ = x^ (1 < z, j < r — 1), under A We are going to prove that A(

Xx E

/

ij-*Ys V V ij E l < t , j < r - l1

+1 XXrrE )j rrH/ rrrr

1<

— =

E

x

X E J^ Vij-&ij iJ +1 ^ r X r ^rrHf r r -rr. 1l < « , j < r - 1l

(5.28)

< i . i'
If £ r r = 0, then (5.28) is true by induction hypothesis. Now assume that xrr ^ 0. Let ^>s\\(

E

xx J / _j ij-*ij-*-ijJij 1< l
±L )) +I XX ±L/ rr rr

rr

—=

- i<*,i<»* E x*-E" l<«,i
v

XijiL/ij,

where x^- = x^t- (1 < z, j < r). Clearly (5.27) and [Ai, A2, • • •, A r _i,x r r ] are of arithmetic distance < r — 1 for all Ai, A2, • • •, Ar_i G F. By (5.14) •4([Ai, A2, ]) — = [Ai, [Ai, A A2, A rA _ jr,_i,x 3?rrj) -A([Ai, A2, •• ••,•,AAr _i,x r _i,xr rr r j) 2, •• ••, •, r r ], therefore X

Ai

ll X

12

^12 X

22 ~~ ^2

X

l,r-1

X

2,r-1

"r x

2r

= 0. *l.r- 1

*l

X

2,r-1 X

*

2r

X* - l , r - l ~~

X*r —t l,r

A,.- 1

X*

«^ rr

1 r—l,r ^rr

Chapter 5. Geometry of Symmetric

240

Matrices

Substitute Ai = 0 and 1 into the above identity, then the difference of the two identities thus obtained gives X

^ 2 2 ~~ ^ 2

x

2,r-1

2r

= 0. .. x*T_- l , r - l

X

2,r-l X2r*

x2r

x X

••.

~

*

X

^r-1

r-l,r

X

rr

rr -- ll ,, rr

Xrr

Proceeding in this way, we obtain I £*_i,r_i — K-i

X

x

r-l}r

Xrr

X

r-lyr

= 0.

Xrr

Substitute Ar_i = 0 and 1 into the above identity, then the difference of the two identities thus obtained gives x*r = xrT. Hence we also have x*_x r — 0. Repeat the above process but with a different order of Ai, A2, • • •, A r _i, we can get x\T = x\r = • • • = x*_2j1. = 0. Therefore Siy A(

E

x J / v XijU/ij ij^ ij l
+

-|- XrrrrJl/rrrr)

=-

+

- E

/ v X—H/ij XrrTTJli , rr^ xljEij -\- XrrlLi < r - ll ll<
where Xij = Xji and x*- = x*-{ (1 < z, j < r — 1). Clearly (5.27) and Z)i<», j
are

adjacent. By induction hypothesis

E

1< l
XijJ^ij)

—

i

i

E

X

ij^Jij->

l
thus the rank of / ^ E

\ ij (««■ X

il < » \ j < r - li

X X

J

ij)*- ij ij)-^ij

~">

XXrrTTJl/ £JTT rr

is 1. We have assumed that xrr ^ 0, therefore x*- — X{j (1 < i < j < r — 1). Hence (5.28) is also true if xrr =fi 0. In a similarly way, we can prove that for k = 1,2, • • •, r — 1

A(
^2 E

XijUfij + + xXkkEkk) == XijEij kkEkk)

^
l < it ,

Yl E i< » *^* r

E x Xij&iji3 + XkkEkk, Xij kkEkk,

j^k j
(5.29) where Xy = x^ (1 < i, j < r, i ^ k or j ^ k). (viii) Consider the image of

E

l < t , i < r - -l

Xijh/ij

+ Zr--l,r(Er-

■l,r

i

-t^r,r--l)

+ X rr^rrt

(5.30)

5.4. Proof of the Fundamental Theorem (I)

241

where x^ = Xji (1 < z, j < r — 1), under A. We are going to prove that ^■4( (

X

^2I X

XijEij ijEij + Xrr-itrtr(Err-itrtr

+ 2?rffr-l) + XrrrrErrrr)

l
= = If xr-i}r

x

x ijEij /5Z j ij-^ij l
(5.31)

+ Xr-itr(Er-i,r + -E^r,r—l) i?r,r-l) T + XrrEXTTrrEJ. TT. i #r—l,rV-^r—l,r T

= 0, then (5.31) is proved to be true in (vii). Now assume that

x

r-i,r 7^ 0. Let

•4 E A{

+

E r. i3 iJ + xx r-lA r-l,r r .-l,r(E

X E Xij-LJij

YJ

-1,T

= Ex*-E--

-0 +

JUyy _£—/*• *• 1 =— +i #J-Jr^rr,r-l) + X rrErr)

1l
^ ^ ^ j , ll<*,j
where *£ = x){ (1 < i, j < r). Clearly (5.30) and £-=i XiEii+xr^r(Er.lir+ Er,r-i) + xTTETT are of arithmetic distance < r — 1. (5.29) with k = 1 implies rr --l l

xXrrrrEErrrr)) A(52\iEu Xr-l,ryEr-l,, • ++ jE? *4(X/ ^ t ^ t t + + ^r-l,r(-Br-l,r ^ r ,r>r r_ - li) ) + + i=l i=l r-l r-l

= 2_^ ]P A { Fit- + + xxr-l,r(E \En _i jr ++ FErri,r-l) r-i) ++JL TTxJZjff TThrr. . r_i ?r (ii/ r- r-l,r zz=i =i

Therefore i i -~ Ai I# #11 ^1 'T* #l,r-2 x Xl,r-2 l,r-1 #l,r-l I

.• • •

. . *T;*.- 2 , r - 2 ** *

x\T

.*

**

Xlr

...

#^ 1l .^r--1l

##1^-2 l,r-2

x

— —

\ r-2 A A r-2

r-2,r-2 X r-2,r-l X r-2,r-l a 'r-2,r X r-2,r

*^lr lr

#Tr*- 2 , r - l ^r_2,r-l

T* X

^X _ -l,r-l ~ A r _i r - l , r - l ~~ "r-1 < - l , r X# r — l , r X r-\,r

~

r-2,r r-2,r #r—l,r #r—l,r X r-l,r ~~ xr-l,r #rr #rr x x I r r ~~ rr

- 00.-

=

x

r-l,r

Repeat the argument of letting Ai, A2, • • •, Ar_i = 0 and 1 successively in some order, which we used in (vii), we obtain x*r = xrr, x*_lr = £7—1,7-, and x\r = • - • = x*_2r — 0. Since (5.30) and (5.27) are of arithmetic distance 2, the rank of >J (X*J — Xij)Eij E (4- ij)t- ij ~r+ XX _i-.i (E.(F-i_i X

ll
J

r

tr

r

)7

r tr r

>r

+ Erfr-i) + Erjr-\)

is 2. It follows that x*j = x^ (1 < i < j < r — 2). Since (5.30) and £i
Chapter 5. Geometry of Symmetric

242

Matrices

of /

0

...

0

. x^r_x - zi, r _i

0

...

0

# r _2, r -l ~

%l r _ l — # l , r — i

. . .

#r_2jT._l

Xr—2,r—1

#r_ijr_i

is equal to 2 for all A € F. It follows that x\r_x

x

r-2,r-l

Xr—ltr—1

# r —l,r

= Xij7._i, • • • ,x*_ 2 r _ 1 =

x r _ 2 ,r-i- Let rank

XijEij = r o

]>^ l
and introduce the notation f =

*'

f 0 if x = 0, \ 1 if* = l.

Then the arithmetic distance between (5.30) and r-2

22 Xi,r-l{Ei,r-l

+ Er-ifi) + £r_i)T._ijEr_i,r_i + Xr_i)T.(Er_i>r + ETir-i)

i=l

is r 0 + 6Xrr. By (5.13), -4(^0 = X for X 6 A--1- Therefore the rank of x

5Z

ijEij + (z*_i,r-i - ar r _i fr _i)E r _i |r _i +

xrrErr

l
is equal to r 0 + &rrr. Hence x*^^

= xr-lfr-i.

(5.31) is now proved.

In a similar way we can prove A(X) = X for X =

x

E

v(Eij

+ Eji) + xkr{Ekr

+ Erk) + xrrErr

(5.32)

l
for any k with 1 < k < r — 1. (ix) Consider the image of r-l

£

x ^

l
+ £

x t r (£ t > + Eri) + xrrErr,

i=r-2

where artJ = a^,- (1 < z, j < r — 1), under A We are going to prove r-l

A(

J2 l
x

ijEij + £ t=r-2

x t >(£, v + JBrt-) + a:rr^rr)
(5.33)

243

5.4. Proof of the Fundamental Theorem (I)

=

r-l

x

iJEH+ E Xir(EiT + Eri) + xrrErr.

J2 l
(5.34)

*=r-2

If £r_2,r = 0 or x r _i >r = 0, then (5.34) is proved to be true in (viii). Thus we assume that xr-2,r ^ 0 and xT-\,r ^ 0. Let r-l

A(

X E Y ^H l
X E Y iri ir t=r-2

+

+ Eri)

+ ^rr^rr) =

Y ^*j^«i> l
where *£ = a $ (1 < i, j < r). Since (5.33) and YZZi A.'£«+Ei=r-2 M ^ r + 2?rt-) + xrrErr are of arithmetic distance < r — 1 and by (5.29) r-l

r-l

A(J2XiEn+ t=l

Y Xir(Eir + Eri) + XrrErr) »=r-2 r-l

r-l = Y liEH «=1

+

^2 Xir(Eir t'=r-2

+ ^rt) +

XrrErr,

we have x?! - Ai L

l,r-2

C

J.r-l

...

L

l,r-2

'r-2,r-2 - A r _ 2 t 'r-2,r-l x x r — 2,r r_2,r

k

''l.r-l

I

•••

lr

l/

X

r-l,r-l

X

r —l,r

Ar_i

r-2,r x T-\,r

*^r—l,r

X--

3r-2,i

= 0.

x

r-l,r

Xr

Repeat the argument used in (vii), we obtain x*r = x r r , x*_lr = a?r_i?r, x *-2,r — #r-2,r? and x\T = • • • = x*_3 r = 0. Since (5.33) and (5.30) are of arithmetic distance 2, the rank of /; (x*j — Xij)Eij l
+ Xr_2,r(£'r-2,r +

E^-2)

is 2. Therefore x*_ 1 | r - 1 = x r _i, r _i, x* jr-1 = x t >_i (1 < z < r — 3), and x*- = xzj (1 < z < j < r — 3). In the above argument, if instead of (5.30) we use / ;

x

ijEij

+ Xr-2,r\Er-2,r

+ ET,T-2) + X r r £/ r r ,

l
then we can prove x*j = x^ for the remaining (z, j) (1 < i < j < r — 1) except (z, j ) = (r — 2, r — 1). Finally, if instead of (5.30) we use (5.27), we will obtain x*_2r_1 = xr-2,r-i- Therefore (5.34) is completely proved.

Chapter 5. Geometry of Symmetric

244

Matrices

(x) Continuing in this way or using an inductive argument, finally we obtain (5.26), but the details will be omitted. The inductive argument is a now completed. Without essential difficulty the proof of Theorem 5.4 for the case when F is of characteristic not two can be generalized to prove the following theorem. Theorem 5.22: Let F and F' be fields of characteristic not two, n and n' be integers > 2. Let A : Sn{F) —> Sn,{F') be a bijective map such that both A and A'1 preserve the adjacency, then n = n\ F and F' are isomorphic, and A is of the form A(X) = atPX<7P + SQ,

(5.35)

where a € F'*, P 6 GLn(F'), a is an isomorphism from F to F ' , and So € Sn(Ff). Conversely, the map defined by (5.35) is a bijective map from Sn(F) to <Sn/(F') and both the map and its inverse preserve the adjacency.

□

5.5

Maximal Sets of Rank 2 (Characteristic Two)

In this section and the next we assume that F is a field of characteristic two and that F ^ F 2 . Definition 5.6: Let F b e a field of characteristic two and F ^ F 2 . A subset C of Sn(F) is called a maximal set of rank 2, if the following conditions are satisfied: 1° C contains a maximal set of rank 1, denoted by M. 2° For any S G C\M there is a unique Ms € M depending on S such that d(5,M s ) = 3 and d(5, M) = 2 for all M ^ Ms in M. 3° For any S € Sn(F) if there is a unique Ms e M depending on S such that d(S9 Ms) = 3 and d(5, M) = 2 for all M ^ Ms in M, then S € C. U

5.5. Maximal Sets of Rank 2 (Characteristic

Two)

245

In this case we also have 1 Proposition 5.13: A maximal set of rank 2 is transformed into a maximal set of rank 2 under a transformation of the form (5.1). □ Define £;, where i = 1, 2, • • •, n, again by (5.7). Then we also have Proposition 5.14: All £;'s, where i = 1, 2, •••, n, are maximal sets of rank 2. Moreover, any maximal set of rank 2 can be transformed under a transformation of the form (5.1) to the set C\. Proof: We prove only the second statement of the proposition. Let £ be a maximal set of rank 2 and M be a maximal set of rank 1 contained in £ . By Proposition 5.8 we can assume that M = Mi. Let S G C\Mi and let Ms be the unique point depending on S in Mi such that d(5, Ms) = 3 and d(5,M) = 2 for all M ^ Ms in M. Write \

V

022/

where sn G F , v is a 1 x (n — 1) matrix, and £22 is a n (n — 1) x (n — 1) symmetric matrix. Since d(5, M) — 2 for all M ^ Ms in A4, rank S22 ^ 2. We distinguish the following cases. (a) £22 is alternate of rank 2. We can assume that S11 — Q(n-3)

Then

Isn S12

5 =

513

^12

SlZ

0 1

1 0 Q(n-3) y

V and there exists exactly one A G F such that

x

sn + A

S12

Sl3

512

0

513

1

1 0

= su + A ^ 0.

We remark that the statement of the following proposition is the same as that of Proposition 5.13 in Section 5.3, so we use the same numbering for this proposition. We do the same also for other propositions and corollaries in this section.

Chapter 5. Geometry of Symmetric

246

Matrices

Since F =fi F2, this is impossible. (b) £22 is non-alternate of rank 2. We can assume that \

S22

S22 —

533

Q(n-3) J

where s22 S33 ^ 0. Then / S n

S12

^12

S13

-S22

«Sl3

533

Q(n-3) y

V and there exists exactly one A £ F such that ^11 + A

$12

512

522

«13

5i3

= (^11 + A) 5 22 533 + 5 33 s\2 + S22 3?3 7^ 0. 533

Since F ^ F 2 , this is also impossible. (c) ^22 is of rank 1. We can assume that

s 22 = ( 5 2 2

0(n"2)

where s22 7^ 0. Then ' $ii

S12

*12

522

0(^-2) Thus for all M G Mu

d(5, Af) < 2, which contradicts 2° of Definition 5.6.

(d) £22 — 0. Then S E C\. This is what we want. We have proved C C C\. Clearly for any S £ Ci\Mi there is a unique Ms £ M1 such that d(5, M 5 ) = 3 and d(5, Af) = 2 for all Af ^ Af5 in A4i. Since £ is a maximal set of rank 2, C\ C £ . Therefore C = C\. □ The proof of the above proposition proves also the following corollaries. Corollary 5.15: Any maximal set of rank 1 is contained in a unique max­ imal set of rank 2. The unique maximal set of rank 2 containing Mi is £1. a

5.5. Maximal Sets of Rank 2 (Characteristic

Two)

247

Corollary 5.16: Any maximal set of rank 2 can be put into the following general form (5.8) / Z11

x12

Zln\

x12

0

0

\xln

0

where P G GLn(F)

P + S0\

Z l l , Z l 2 , * - ,Xln

G F >,

0 I

and S0 G Sn(F).

D

Propositions 5.17 and 5.18 of Section 5.3 hold also for the case when F is a field of characteristic two and F ^ F 2 , and the same proofs apply. For completeness we list them as follows. Proposition 5.17: Let £ be a maximal set of rank 2 and So G £ , then there is one and only one maximal set of rank 1 contained in £ and containing So. □ Proposition 5.18: Any two distinct maximal sets of rank 2 which have a nonempty intersection can be carried simultaneously by a transformation of the form (5.1) to C\ and £ 2 . □ However, Proposition 5.19 in Section 5.3 does not hold for the case when F is of characteristic two. Let

Cij(t)= Xl + tEiAbil

+ tEij),

where t G F. Clearly, £ t j(0) = £ t . Instead of Proposition 5.19, we have 2 Proposition 5.23: Let F be a field of characteristic two (and F ^ F 2 ). Then for 1 < i < j < n, dj(t)'s, where t runs through F , and Cj are all the distinct maximal sets of rank 2 which contain 0 and E^ + Eji in common. Proof: It is enough to consider the case (z, j) = (1, 2). Following the proof of Proposition 5.19 in Section 5.3 we conclude that £ i , £ 2 and £i 2 (£), where t G F, are all the maximal sets of rank 2 which contain 0 and E\2 + E2\ in common. We have £i 2 (0) = C\. Clearly, £ 2 and Ci2{t)fs, where t runs through F , are distinct in pairs. Q 2

The parentheses in Propositions 5.23 - 5.25 indicate that these propositions hold also for F — F2 after we define maximal set of rank 2 in Section 5.7.

Chapter 5. Geometry of Symmetric

248

Matrices

Proposition 5.23 implies the following more general result. Proposition 5.24: Let F be a field of characteristic two (and F =fi F 2 ). Let S be any n x n alternate matrix of rank 2. Then there exist exactly | F | + 1 distinct maximal sets of rank 2 which contain 0 and S in common. □ On the other hand we have Proposition 5.25: Let F b e a field of characteristic two (and F ^ F2). Let So be an n x n non-alternate symmetric matrix of rank 2 and assume that both 0 and So are contained in a maximal set of rank 2 simultaneously. Then this maximal set of rank 2 is the only maximal set of rank 2 containing both 0 and So. Proof: By Proposition 5.14 we can assume that both 0 and So are contained in C\ and So is of the form (s n (»n \\ 1 1 0 2 V(l (»-2)> f/ 0o("where Su ^ 0. Let £ be a maximal set of rank 2 which contains both 0 and So- By Proposition 5.17 there is a unique maximal set of rank 1 contained in C and containing 0, let it be called M. Then every element distinct from 0 in M is of rank 1 and S 0 G C\M. Since F ^ F 2 , by 2° of Definition 5.6 there is an S G M S ^ 0, and d(5 0 , S) = 2. Write S in the block form S = So0 =

Sl2\ = ( ^u ^u "\ S = S12 S22J £22 / '' ' \VS12 where S u is a 2 x 2 symmetric matrix, S12 is a 2 x (n — 2) matrix, and 522 is an (n — 2) x (n — 2) symmetric matrix. Since r a n k S = 1, we have r a n k S u ^ 1- We distinguish the following two cases. (a) S u = 0. Since r a n k S = 1, we have Si 2 = 0. Then r a n k S ^ = 1 and So - S is of rank 3, i.e., ad(S 0 ,S) = 3. Thus d ( 5 0 , 5 ) = 3. This is a contradiction. Hence Su = 0 cannot occur. (b) Sn 7^ 0. Since S u is a 2 x 2 symmetric matrix of rank 1, there are only three possibilities:

SU:

■C a)- (°

.)•

(;

s-H2)-

5.5. Maximal Sets of Rank 2 (Characteristic Two)

249

In the first two cases s ^ 0 and in the third case st ^ 0. For the first case, from rank S = 1 and d(S0,S) = 2 we deduce 5 = sEu, then by Corollary 5.9 M = Mi and by Corollary 5.15 C = C\. For the second case we have S = sE22, then M = M2 and C = C2. But C2 does not contain 5 0 , therefore the second case cannot occur. For the third case, by Corollary 5.9, we can assume 3 = 1. From rank 5 = 1 and d(5 0 , S) = 2 we deduce

(I S=\tt2 \

t

\ 0/

Let P = I + tEw, then the transformation X h—* *PXP carries S into En. By Corollary 5.9 there is a unique maximal set of rank 1 containing 0 and £ u , which is Mi, and by Corollary 5.15 there is a unique maximal set of rank 2 containing Mi, which is C\. It follows that M = *PMiP and C = *P£iP. Then C consists of elements of the form / Zn I ^Xn + £12

\

tXu + X12 ^2^11

X13 ^13

• • • Xin \ * * • tXin I

x13

txi3

0

•••

Xin

tXin

0

•••

0 0

I /

where i n , Xi2, ^13, • • •, Xin are arbitrary elements of F. Clearly, ^o is not an element of the above form. Therefore the third case cannot occur either. Hence the first case is the only possible case which gives us C = C\. Q Propositions 5.24 and 5.25 distinguish the alternate matrices of rank 2 and the non-alternate matrices of rank 2 contained in a maximal set of rank 2 containing 0. We introduce the following notations. Let £ be a maximal set of rank 2 containing 0. Denote by C^ the set of alternate matrices contained in C and by C^ the set of non-alternate matrices contained in C. For example,

Chapter 5. Geometry of Symmetric

250

Matrices

for C\ we have

(°

0)

4 =< k

* *

Xln\

Z12

0

•

0

V^ln

0

• •

0 /

#12

\ ,xine

£l2,' •

F\ )

and r

n)

/^n a; 12

4 =< k

V^ln

X12

0 0

• •

Xln\

•

0

• •

> xu 6 F\

£ i 2 , - - * ,Xln

G

JF

► .

^

0 /

Now let us study the intersection of two maximal sets of rank 2. Let C and C be two distinct maximal sets of rank 2 and assume that CC\ C ^
M" =

KJ^r

where Px G GLn^(F)

A

*' do"'

and A G F. By Corollary 5.15

t

£" =

A42

I(n 2)

~)

fc ,r -('vJKr-

Denote the first row of i \ * by (p2, • • • ,p n ). Then (p 2 , • • • ,p n ) ^ 0 and

/

0

X12P2 cAx n £" £" == J< | ~y \ Xl2Pn

Therefore we have

X12P2

0 0

• •

Xl2Pn \

0

. ••

0

;

1

>

11 2Xl2€ F e F> . Ui /

5.5. Maximal Sets of Rank 2 (Characteristic

Two)

251

Proposition 5.26: Let F be a field of characteristic two and F / F 2 . Let £ be a maximal set of rank 2 and assume that C\ ^ £ and C\ H £ ^ <£. Then £ i 0 £ is of the form f

An£'=<

X

Ik

/r 0U P2 P2 * P2 0 • ••

\Pn \Pn

0

•••• ••

0

+ So x e F

► »

OJ 0/

where p 2 , • * *, Pn G F, (p 2 , • • ■ ,p n ) 7^ (0, • • •, 0), and #0 is an element of

A n £.

n

Corollary 5.27: Let F be a field of characteristic two and F ^ F 2 . Let £ be a maximal set of rank 2 and assume that 4 a ) H £ ' 7^ f

/o £[a) n£'= <

P2

X

k

\Pn

P2

' •*

0 • •• 0 •

Then

>

Pn\

0

+ S0 x e F

►

,

.. oJ

where p 2 , • • •, p n G F, (p 2 , • • • ,p n ) ^ (0, • • •, 0), and S0 is an alternate matrix of C[a) H £'. D

Definition 5.7: Let F be a field of characteristic two, F ^ F 2 , and n > 3. Let £ be a maximal set of rank 2 containing 0. A line in C^ is the intersection of C^ with another maximal set £ of rank 2 such that £(°) f]

From Corollary 5.27 and Definition 5.7 we have the following proposition. Proposition 5.28: Let F be a field of characteristic two, F / F 2 , and n > 3. Let £ be a maximal set of rank 2 containing 0, then C^ has an (n — l)-dimensional affine space structure over F . More precisely, let C be the set (5.8) with So = 0, then the map £(a) ^ A G ( n - l , Ff ) £<•> _--v

/ 0 t

p

X12

Vrcln

Z12

0 0

•

• • •

0 0 )

P

■—» K - • ,

^ln)

Chapter 5. Geometry of Symmetric

252

is a bijective map which carries lines in C^ dimensional affine space AG(n — 1, F).

5.6

Matrices

into lines in the (n — 1)□

Proof of the Fundamental Theorem (II)

Proof of Theorem 5.4 (the Case F Being of Characteristic Two and ^ F 2 ): Let F be a field of characteristic two and F / F 2 . We prove only the first statement of Theorem 5.4. Consider first the case n > 3. We proceed in steps. (i) Let A : Sn(F) —-> Sn(F) be a bijective map and assume that both A and A'1 preserve the adjacency. By Definition 5.2 A preserves also the distance. As in Section 5.4 we can assume that (5.9)

.4(0) ==0,0, A(0)-that

A(Mt) A(Mi) --== Mui ,

(5.36)

A(C1) --== CA. A{&) 1.

(5.37)

and that By Propositions 5.24 and 5.25 we have also a) A{C[a)a))) -== C[ A(£[ 4a)

and

n) n) A(£[ 4(4 n) ) ==44n ) . .

(5.38)

Clearly, A induces a bijective map from £{* to itself which carries lines into lines. Let

A

(°

X12

X12

0

• •

0

/ 0 2/12 2/l2

V^ln

0

• •

0 )

\yin

•

(°

2/12 *••• 2/12 ••

yln\ Vln\

0

•• ••• • 00

0

•••• ••

0J

0 /

Then by Proposition 5.28, the assumption F ^ F 2 , the fundamental theorem of the affine geometry, and A(0) = 0, we have (2/12,- • ? 2/ln) = (ffl2, ' •

5 %ln)

-L-)

5.6. Proof of the Fundamental Theorem (II)

253

where a is an automorphism of F and P G GLn-i(F). to the bijective map of Sn(F)

-M1

Vi

After subjecting A

P-

we can assume that for all X e 4 a ) -

A(X)=X

(5.39)

(ii) Since A(Ci) = C\ and A(Eu + En) = Eu + En (i ^ 1), by Proposition 5.23 we must have A(d)

= Ai(*t), i = 2, 3, • • •, ri,

where tt- G F . After subjecting .4 to the bijective map of

F . - + < ( / + £ tiEa)Y(I

Sn(F)

+ £ UEn),

*=2

t=2

we can assume further that A(d)

(5.40)

= CU i = 2, 3, • • •, n.

Consequently, A(d

fl £ , ) = d ClCj, l
n.

In particular, A(Eij + Eji) = Xij(Eij + Eji)

for some Xij G F*.

(5.41)

From (5.40) we deduce A{C{2a)) = C(2a). Let /

0 \

0

X12

0

^12

0

Z23

^2n

0

z23

0

0

/ 0

y12

0

M/12

0

?/23

y23

0

I= I 0

V o x2n o • • • o /

0 \ 0

v o y2n o . • • o /

By the fundamental theorem of the affine geometry and *4(0) = 0, we have (2/12, 2/23, ' * * , 2/2n) = 0*12, ^23, * * * , ^ n ) *

2

^

(5.42)

Chapter 5. Geometry of Symmetric

254

Matrices

where a2 is an automorphism of F and P2 € GLn-i(F). Substituting' (1, 0, • • •, 0), (0, 1, 0, • • •, 0), • • •, (0, 0, • • •, 0, 1) successively into (5.42), we deduce from (5.39) and (5.41) that P2 is a diagonal matrix of the form [1, p 3 , • • •, p n ], where p 3 , • • •, pn ^ 0. Substituting (x12, 0, • • •, 0) into (5.42) we deduce from (5.39) that cr2 = 1. Therefore / 0

X12

0

Xl2

0

X23

•••

0

x23

0

V 0

x2n

0

x12

0

^2n

0 \

/ 0 ^12

0

£23p3

^2nPn

0

0

x23p3

0

0

X 2n Pn

0

0 /

\ 0

•••

..

0

\

o /

(5.43) We assert that p3 = • • • = pn. If n = 3, there is nothing to prove. Consider the case n > 4. Clearly

d(£(£„ + En), J2(E2i + Ei2)) = 3. «=3

i=3

Since A preserves the distance, by (5.39) and (5.43) we have

d(J2(Eu + En), X > ( £ * + Ei2)) = 3. t=3

t=3

Therefore p3 = • • • = pn. After subjecting A to the bijective map of Y,

-^P3(1

ft1/<-l>)y(1

Sn(F)

ft1/^1))'

which leaves every element of C[ fixed and every £ t (i = 1,2, • • •, n) invari­ ant, we can assume that (5.9), (5.37), (5.38), (5.39), (5.40), and A(X) = X for X € 4 a )

(5.44)

hold. By Proposition 5.17 and from (5.9), (5.37) and (5.40) we deduce A(Mi)

= Mi,

i = 1, 2, • • •, n.

(5.45)

(iii) By (5.38) we can assume that for Xu £ F*, £i 2 , • • •, #i n £ F , xu

0

0

Vxi n

0

0 /

xn

x12

^12

\*i»

o

0

o /

5.6. Proof of the Fundamental Theorem (II)

255

where x*xE F*, x\2, • • •, x\n G F. Clearly, /Zll

#12

x12

^ln\

0

and

0 \Xin

/ 0 x12

x12 0

0 \xln

0

£ln\

0

0 /

are adjacent, thus /x* x\2

0

\*i»

0

0 / / 0

*ln\

12

0 ...

0

\xm

0

and

0 /

are adjacent. It follows that x\2 = #12, / Z11

x12

Xln\

xu

0

0

x12

Zl2

Xln\

0 * ...

, x\n = xin. / #u

^12

^12

0 /

Therefore Xln\

0

0 \xln

0

0 /

\xln

0

0 / for all £11 £ F*, £12, • • *, £i n € F . By (5.45) with i = 1 we can assume that

^ C " o(«-)) = ("n o(«-))

foralll er

"

Since the distance between / £11

£12

£12

^ln\

0

0 \xln

and

( I u o^-1))

and

\

0

is 3 and the distance between

0 )

/ x\x x12

x12 0

0

\xln

0

0 )

o^-1))

is 2 or 3 when x\x ^ xn or a ^ = i n , respectively, we must have x^ = X\\. Therefore x\x depends only on xn and does not depend on #12, #13, • • •, X\n.

Chapter 5. Geometry of Symmetric

256

Matrices

Similarly, we can prove that 0 \

/ 0

x12

0

X\2

X22

#23

X2n

0

X23

0

0

V 0

Z 2n

0

I

=

0 /

•

/ 0

X12

0

I #12

#22

^23

X2n

0

z23

0

0

V0

x2n

0

0 /

0 \

where x*2\ depends only on x22. It is clear that for x =^ 0, the distance between Ix 0 x \

0 x 0 0 0 0

\

/0 and

Q(n-3) y

X

0

Q(n-3) y

V

is 2. Then the distance between their images under A is also 2. Thus X* 0 x 0 x** x = 0, x x 0 from which we deduce x** = z*. For x ^ 0, r

r

x 0 0 0

and 0(n-2) y

^

0 x

x x Q(n-2)

are adjacent, thus their images under A are adjacent. Therefore x

x

X

X*

= 0,

from which we deduce x* = x. Hence A{X) = X

for all X G £1 and £ 2 .

We can prove further that A(X) = X

for all X € A , * = 1, 2, • - •, n.

(5.46)

5.6. Proof of the Fundamental Theorem (II)

257

For i = 1 or 2, this is true. Now consider the case i > 3. We consider only the case i = 3, since the other cases can be treated in a similar way. From (5.40) we deduce . 4 ( 4 a ) ) = 4 * 0 - Let / 0 0

0 0

#13 X23

0 0

Zl3

X23

0

Z34

£371

0

0

Z34

0

0

V0

0

x3n

0

•••

0 0

2/l3 2/23

0 0

^13

2/23

0

2/34

2/3n

0

0

2/34

0

0

V0

0

y3n

0

/ 0 0

0 \ 0

0 /

0 \ 0

•••

0 /

By the fundamental theorem of the affine geometry and .4(0) = 0, we have (2/13, 2/23, 2/34, • • • , 2/3n) = (^13, X23, *34, ' ' * , X^n)"3

P3,

where 03 is an automorphism of F and P3 € GLn-1(F). As in Step (ii) we can prove that P3 is a diagonal matrix of the form [1,1,P4, • • • ,^4], where P4 T^ 0, and 03 = 1. Therefore / 0 0 Zl3

0 0

x13 z23

0 0

£23

0

£34

X3n

Z13

0 V 0

0 \ 0

0

0

I34

0

0

V0

0

a;3„

0

0 /

for X13, a;23, X34, • • •, X3„ € F. xE33 for all x G F .

/ 0 0

0 0

313 123

0 0

£23

0

X34JJ4

X3nP4

0

x 34 p 4

0

0

0

Z3nP4

0

0

By (5.45) we can assume that A(xE33)

For x ^ 0,

/o

/0 a; x x 0

V

and Q(n-3) J

V

Q(n-3) J

are adjacent. Then their images under A, /0

/o

\ a; x x 0

0 0

and Q(n-3) J

Q(»-3) )

\

/ —

Chapter 5. Geometry of Symmetric

258

Matrices

are adjacent. Therefore x = x. As in Step (iii) we can deduce / 0 0

0 0

xi3 x23

0 0

#13

#23

#33

#34

0

0

x34

0

\ 0

••• •••

0 \ 0

* * * #3n

•••

0

0 # 3 n 0 - " 0 /

0

0

0 0

#13

#23

#33

#34^4

0

0

#34P4

0

\ 0

0

X3nP4

0

/

#13

0

..

#23

0

...

o \ 0

' * ' #3nP4

...

0 0

/

for a:13, x 23 , ^33, #34, * ■ *, #3n G F. Clearly, d(jEn + E\4 + £41, ^33 + ^34 + £43) = 2. We have A{EU

+ E14 + E41) = En + E14 + E41, A(E33 + E34 + E43) =

E33 + p4{E34 + E43). Therefore d(Eu + E14 + E41, E33 + p4{E34 + E43)) = 2. It follows that p4 = 1. Hence (5.46) holds for i = 3. Now we have arrived at the main conclusion of Step (iii) of the proof of Theorem 5.4 for the case when F is of characteristic not two in Section 5.4, i.e., we have (5.46). We may proceed Steps (iv) - (ix) of that proof and finally we will arrive at the conclusion A(X) = X

for all X €

Sn{F).

Therefore Theorem 5.4 is proved for the case when F is of characteristic two, F ^ F2, and n > 3, but we have to make the following remarks on that proof. Remark 5.1: Case (a) in Step (iv) needs some modification. In the present case, i.e., the case when F is of characteristic two and F ^ F 2 , we have also (5.15) and (5.18). From (5.18) we deduce x^ = Ai and x\2 — A2 immediately, then from (5.15) we deduce x\2 = 0. Remark 5.2: In Steps (v) - (ix) that A preserves the arithmetic distance is used. However, this can be easily replaced by the fact that A preserves the distance. In fact, we have assumed that F ^ F 2 , then there are at least four possible choices for each At € F or x G F and the arithmetic distances

5.6. Proof of the Fundamental Theorem (II)

259

appeared in these steps can be replaced by the distances. For instance, in Step (v), we may choose Ai G F with Ai ^ xu and rc^, then both r

A £E X ]/ T> XijEij ij&ij - ^2 ^i "

- Et = l ' «

l
t=i

and A x*-E-xljEij-j^XiE*

E E

-E .^»

l
7,

t'=i t=l

are non-alternate, thus by Proposition 5.4 r r XXijU/ij) A E j:\iEu)d( J2 HEih X! * »)==a dad( ( Ys£ XXij&ij, i3Ei3, DA) Y2XiE») d( E l
t=l

l<*\j
t'=l

and xTjEij, E AJ:\iEii). ^«). £t = l W =:ad(l <x: ij 31 t*,\jj<
X E d( *j Hi d( £E *'&,

£ A , - £ « ) = ad( £

l < t , j
x Ei

n

t=i

t=l

Since XijU/ij^ *ijEih ! £ E x*-E-4^;> EW E W >£W « ) ===d(<*( E d(l < £tE t'=i ,j
ll
«=1

t=l

and

ad( £

x

ij^ij-)

r i^Eu)


=i \j
we have

rank(

^

x* E

i<«*,i
rank( £

A E --^) < r t'=i r

z*^-, £ A t £ t t ) < r

i<«*,i
t=i

for Ai ^ Xn, x\x in F and any A2, • • •, Ar G F , from which we deduce also x*st = 0 for r < 6, t < n and :r*t = 0 for 1 < s < r < t < n. Now consider the case n = 2. As in the case n > 3 we can assume that A(0) ,4(0) === 00 and A(Mi) == Mi, M.-, ^(M)

i = l1,2. ,2.

We may write 'a ^(

\

0\

v° ° / ~

=fVfln0 S) 0)

for all a£ F

260

Chapter 5. Geometry of Symmetric

Matrices

and * ( !

for all a G F ,

«j (o

«')

r n n i T T* where a and a' are bijective maps from F tor-v iitself with (T = (T = 0. After subjecting A to the bijective map

X —+ X-» ( l * (l^X )"1* we can assume further that 1^ = 1. From -4(0) = 0 we deduce immediately that A carries matrices of rank 1 to matrices of rank 1, non-alternate matrices of rank 2 to non-alternate matrices of rank 2, and nonzero alternate matrices to nonzero alternate matrices. Therefore we can write

°T cTo)'

4" i - U C

) =(

A(°

\c

{ c

0)

0

where r is a bijective map from F to itself with 0T = 0. From A(Mi)

= Mi we deduce A(£i) = £,, * = 1,2. Consider the image of

M U oj faa

b b 0

under A where a, 6 G F * . F r o m d ( ( ^ ) , ( ° »)) = l a n d « * ( ( ; $ ) , ( ; $ ) ) = 3 we deduce " ( k6

OJ-■(i o)

* '0 1 A \ ,6

fc\ =

for all a,&£ F*.

Similarly,

dj

=

/ 0

6T \

{ V d"' ) for all

b,d£F*.

For any a e F*, from the adjacency of a 0\

v° % 0° 0; J (n

n)

, / 0 a and and

(° y [a aM aaJ

we deduce r 2
(5.47)

5.6. Proof of the Fundamental Theorem (II)

261

Substituting a = 1 in the above equation, we obtain V' = ( l r ) 2 . Therefore after subjecting A to the bijective map X x —- ( (

(r)( I 1T 1))X* !(

( I(nT1

1

)

we can assume further that V' = 1. For a e * - , let , * ( ; °) = ( * * ) • Since ( ° ° ) is adjacent with ( ° ° ) , (J» ° ) , and f j j ) , we have xz + xz + a? a^zz == yy2,2, xa aa' +, xz = y\2 xa + xz = y , and xz XZ a

2 = (a r ) 2 . = yy2 + (a

T-

a

It follows that x = a , z = a , and y = 0. Therefore 'a * ( ,0

0) a)

1 =( »1

for all a G F*. «")

Consider the image of ( j * ) under A, where a,6 € F*. From <*((**), ( ; j | ) ) = 3 and «*((;*) , ( • » ) ) = 2 we deduce ( a" * ( >*

a

for all a,be

F*.

- )

)"

Clearly, this equation also holds when a = b — 0 or one of a and b is zero. Let a,d £ F* and a ^ d. Assume that

A\ a A

*

k

bU b

Since (^)

is a d

Jacent

T u 'V b\

( x

■ ( " , 7 1 - 1

<
a I

\ y

y

*/ z

witn botn

) a n d ( j j ) , Tw2 e have (x + a ' ) ( z(+j j«.*') =~-(y + b ) (x + a')(z + a*') = (t/ + 6T)2

and

a°)z-- (y + bTT)22(x + + O z == (s/ +6).

(x

262

Chapter 5. Geometry of Symmetric

Matrices

It follows that x = a". Similarly, z = d" . Therefore y = bT. Hence A

.(a \ b

bT d"'

b\_(a° d )~\bT

Clearly, the above equation also holds when a = d and when a = 0 or d = 0. Therefore we have A

\b

a b\

( a'

d) = \v

V

#)

fo^lla,MGF.

In particular

^:\)-{{7'i)

*-•"-'•

Since ( aa * ) is of rank 1, we have (a2y = (a T ) 2

for all a G F.

(a2)*' = (a r ) 2

for all a G F.

Similarly,

From (5.47) we deduce

(a*y(ay = ((a*yf. From the above three equations we deduce

(«T)2 = («TTherefore

(«2r = («T' = («TMoreover, we have A l

a2 ab ab b2

\_((a2y )-{(aby

(aby (by

For a ^ O , (f6 $) is of rank 1, from which we deduce (a2y(b2y' But (a2)' = (aT)2, (b2y' = (bT)2. So we have

(ay(¥f

= ((aby)2,

= ((aby)2

5.6. Proof of the Fundamental Theorem (II)

263

and then {ab)T = aTb\ which holds evidently also for a = 0. Assume that a + b ^ 0. From the adjacency of (alba^b) deduce

and (bQa\)

(a° + b")(a + by' = ((a + by)2. By (5.47)

((a + by)2 = (a + bY(a + by'. Consequently, (a + b)a = aa + 6", which holds for all a, b € F . Similarly, (a + &)"' = a a ' + b°'

for all a,b e F.

We compute ((a + 6) T ) 2 =

((a + &)2)T = ((a + &) 2 r = (a 2 + & 2 r 2 2 2 2 2

= (a y + (b y = (a y + (b y = (
((a + byy = (a + by (a + by' a" a"' + aab
= But ((a + by)2

= (aT + bT)2 = (aT)2 + (F)2 =

a" a"' + b"b"'.

Therefore a°b°' = b°a°'.

we

264

Chapter 5. Geometry of Symmetric

Matrices

For b = 1 we obtain a? = aa' for all a 6 F , i.e., a = cr'. Then (5.47) becomes (a*) 2 = (a T ) 2 . Therefore a a = aT for all a e F, i.e., cr = r. Hence ^4 is of the form ( a" b" \ ' a &\ for all a,b,d€ F. " \ b" d° ) *( Now Theorem 5.4 is also proved for the case when F is of characteristic two, F ^ F 2 , a n d r a = 2. □ Notice that the proof of the case n — 2 holds also when F = F 2 .

5.7

Proof of the Fundamental Theorem (III)

We begin with the following lemmas which are of independent interest. In Lemmas 5.29 and 5.30 we assume that F = ¥q is a finite field with q elements where q is a power of 2. L e m m a 5.29: Let S be an n x n alternate matrix of rank 2s over F g , where q is a power of 2. Denote by n2s+i(S), where i = 0, 1, 2, the number of points W e Sn(¥q) for which d(0, W) = 1 and d(W, S) = 2s + i. Then n2s(S) = q2s - 1, n2s+1{S) = qn - q2% and n2s+2(S) = 0. Proof: Without loss of generality we can assume that

S S =

(/ /W °0

//W « 0

°

\ Q(n-25) y

Let W e Sn(¥q) be such that d(0, W) = 1 and d(W, S) = 2s + i. From d(0, W) = 1 we deduce ad(0, W) = 1, i.e., mnkW = 1. Thus W can be expressed as

W W = *(«!» *(au «2, a 2 , •' • ••»•,« nan) )(a ( a iu,

a 2a,2 ,•••">• •, O na) ,n ),

where (01, a 2 , • • •, an) ^ 0. We distinguish the following two cases, (a) (a 2s +i, a 2 s + 2 , • • •, an) = 0. Then (au a 2 , • • •, a 2s ) ^ 0 and W = *(ai, a 2 , •• 0)(ai, • •* •,5 a«25 0)(ai,a 2 ,a 2•, ••*,• a•,2s ,a 2s • • •, 0). 2 5,, 0, • •, •• •, 0,, •0,••,0).

(5.48)

5.7. Proof of the Fundamental Theorem

(III)

265

Thus _ ( *(ai, «2, • • •, a2s)(au

s

W

a 2 , • • •, a2s) + lj{s)

Q

\

)

\ Q(n-2s)

)

is a non-alternate matrix. It can be easily computed that / 0 j(s)' a 2 , • • •, a 2s ) + f / ( s ) Q

/ det ^ *(au a2, • • •, a2s)(au

= 1.

Therefore there are q2s — 1 points VK of the form (5.48) with (a 2s +i, a2«+2, • • •, a n ) = 0 for which d(0, W) = 1 and d(W, 5) = 2s. (b) (a2s+i, G2s+2, • • •, a n ) ^ 0. There is a Pi E G r L n _ 2s (F g ) such that (a 2s +i, a2s+2, • * *, an)Pi = (1, 0, ■ • •, 0). Let

_ (IK

]

\

then *PWP = \au

a2, • • •, a 2s , 1, 0, ■ • •, 0)(ai, a 2 , • • •, a 2s , 1, 0, • • •, 0)

and l

PSP = S.

Thus *P(W I

S)P 0 1^ (au a 2 , • • •, a25, l)(ai, a 2 , • • •, a2s, 1) + I /(*) 0

t

\

Q(n-2«-l) J

It can be easily computed that / / 0 /(5) {s) det I *(ai, a 2 , • • •, a 2s , l)(ai, a 2 , • • •, a25, 1) + I 0

11=1.

Therefore there are qn—q2s points W of the form (5.48) with (a2s+i, a2s+2, • • •, an) ^ 0 such that d(0, W) = 1 and d(W, 5) = 2s + 1.

266

Chapter 5. Geometry of Symmetric

Matrices

Assembling the above two cases, we get the conclusions of our lemma.

□

L e m m a 5.30: Let S be an n x n non-alternate matrix of rank 23 + 1 over ¥q, where q is a power of 2 and s > 0. Denote by n2s+i(S), where i — 0, 1, 2, the number of points W € Sn(¥q) for which d(0, W) = 1 and d(W, S) = 2s + i. Then n2s{S) = (?2s - 1, n 2 5 + 1 (5) = (?2s+1 - (?2s, and n 2 s + 2 (5) = qn - q2s+1. Proof: Without loss of generality we can assume that / j(2»+l)

s =v

n 25_1 Q(n-J

j• o( ->•-!) ))'

Let W £ Sn{¥q) be such that d(0, W) = 1 and d(W, 5) = 2s + i. As in Lemma 5.29 we also have rank W = 1 and VK can be expressed as (5.48) ( aui , aa22,, •• ••"•,j aann)),, W == *(ai, *(ai, aa22,, •• • • •,•, aann)){a

where (ai, a 2 , • • •, an) ^ 0. We distinguish the following two cases. (a) (a 2 5 + 2 , a 25+3 , • • •, an) = 0. Then (ai, a 2 , • • •, a2s+i) ^ 0 and W = *(ai, a 2 , •• ) ( a ui , aa22,, •• ••'•,, «25+l, • • *•,, ^25+1, a 25 +i, 0, 0, •• •••,•, 00)(a a 2 a + i, 0, 0, •• ••■•,, 00). ).

Thus a 25+1 2 • • •, wr ,+i)(ai, a<J2) ) 2 , • *• •, a2s+i) + /( w-s=(c _ ( \ (ua i '°<*2, ' "■• > a 2fl2s+l)(ai,

\

Q(n-2s-l) J •

It is easily computed that 2 + 11 ) det ( *(ax , a 2 , •• • •, a 2 s,+i)(ai, + i ) ( a i , a 2 , •• •••, ))) - , aa2255 ++i) i) + + JO / ( 2 s'*

= l + a2 + a2 + --- + a25+1. Consider first the case 1 + a\ + a\ + h a 2s +i = °- Clearly the number of nonzero (au a 2 , • • •, a 25+1 ) for which 1 + aj + a\ + • • • + a 2 s + 1 = 0 is equal to q2s. Since s > 0 and (en, a 2 , • • •, a 2s +i) ^ 0, there is at least a j (1 <j < 25 + 1) such that l + a 2 + a 2 + - - + a2j_1 + a | + 1 + -•- + a 2 s + 1 ^ 0, thus rank(W - 5) = 2s. If ax = a2 = • • • = a 2s +i = 1, then VF - S is an alternate matrix and d(W, S) = 2s + 1. If (a x , a2, • • •, a 2 a + 1 ) ^ (1, 1, • • •, 1), then W -S is non-alternate and, hence, d(W, S) = 2s. Therefore among the

5.7. Proof of the Fundamental Theorem

267

(III)

points W of the form (5.48) with (a25+2? «2s+3, ■ * *, «n) = 0 and 1 + af + a?> H h aL+i = 0? there is exactly one point W such that d(0, W) = 1 and d(W, 5) = 2s + 1, and there are g 2s - 1 points W such that d(0, W) = 1 and d(W, 5) = 2s. Then consider the case 1 + a\ + CL\ + • • • + ^Is+i 7^ 0* Clearly the number of nonzero (ai, a 2 , • • •, a 2s +i) for which 1 + a\ + a\ + • • • + a\s+1 ^ 0 is equal to q2s+1 - q2s - 1. Therefore there are g 25+1 - q2s - 1 points W of the form (5.48) with (a25+2, «25+3, • ■ *, o,n) = 0 and 1 + a\ + a\ -\ h a 2 5+1 ^ 0 such that d(0, W) = 1 and d(W, S) = 2s + 1. (b) (a 2s +2,fl2s+3?•••, «n) 7^ 0. Then n > 2s + 1 and there is a P 2 € (j?Ln_2s-i(Fg) such that ) P 2 == (l, (a2s+2, (a a 2 s + 3 , -"- • ,>a aBn)-p2 ( 1 , 0, 0, ••■ •,o). • •, 0). 2 5 +2, 02s+3,

Let

,

-r" «)• x

//(2H-1)

P--

*)•

then •, 0)(ai, a 2 , •• * 5 «2s+l 5 1 5 0, • -•, o ) ? «2s+l, 1) 0, • ■ *PWP = *(ai, a 2 , • * *, G23+1, 1, 0, • • •, 0)(ai, a 2 , • • •, a 2 ,+i, 1, 0, • • •, 0) and

W P = *(ai, a2, •• *

'PSP = S. Thus

*P(W - S)P

(

/
(*(<*!, « 2 , •'

-1

X

•, a 2 5 +i, l ) ( a i , «2, •• ' , «25, 1) + f

Q(n-2s-2)J Q(TI-2S-2)J * ( a i , a 2 , • • •, 025+1, l ) ( a i , <*2, • • •, ^2*, 1) + (

Q

)

is non-alternate. It is easily computed that /

//j(2s+l) /(2s+l)

a det f( £*(«!, (ai, a 22, • •* •,> a«2s+l +i, 1) + ( l ) ( a l > «2, 2 , *•,'■ a•,2 5 «2 2 s + i,? l)(«i, + i,5 1 ) + (

x\

J ) = 1. 0 )Q ) -

Therefore there are qn — q2s+1 points W of the form (5.48) with (a 2s +2, «2s+35 • • •, an) ^ 0 such that d(0, W) = 1 and d(W, 5) = 2s + 2.

268

Chapter 5. Geometry of Symmetric

Assembling the above cases, we get the conclusions of our lemma.

Matrices □

Remark 5.3: In Lemma 5.30 when s = 0, i.e., S is a 1 x 1 matrix of rank 1, we have n2.o(S) = 1, n2.o+i(5') = q — 2, and n2.o+2(S) = 0.

□

The following lemma is crucial in the proof of Theorem 5.4 for the case F = F 2 and n ^ 3. L e m m a 5.31: Let n be an integer > 2, n ^ 3, and A : <Sn(F2) —► <5n(F2) be a bijective map. Assume that both A and A'1 preserve the adjacency. Then A preserves the arithmetic distance. Proof: We want to prove that for any S,T

£ <Sn(F2),

■ ad(5,T) = a d ( ^ ( 5 ) M ( r ) ) . Clearly, we can assume that S ^ T. For a fixed pair 5 , T G <Sn(F2) with S ^ T, define a map A1:Sn(F2)

—»

S„(F 2 )

x •—► ^(x + r ) - ^ ( T ) . Clearly, both .4i and A±l preserve the adjacency. Then Ai preserves the distance. We have A (0) = 0 and A^S

-T)

= A{S) -

A{T).

Thus d(0, S-T)

= d(Ai(0),

AiiS - T)) = d(0, A(S) -

A(T)).

If S — T and A(S) — A(T) are both alternate or both non-alternate, then by Proposition 5.5 we deduce ad(0, S-T)

= ad(0, A(S) -

A(T)).

Consequently, ad(5, T) = ad(A(S),

A{T)).

5.7. Proof of the Fundamental Theorem

(III)

269

Then consider the case when only one of S — T and A(S) — A(T) is alternate. For definiteness, assume that S — T is alternate and A(S) — A(T) is nonalternate. Let rank(5' — T) = 25, where 5 > 0, and rank(^4(S') — A(T)) = r. By Proposition 5.5, - T) ++ 11 = 2s T) == ad(0, ad(0, SS-T) 25 + 1, d(0, S --- T)

A(T)) = ad(0, A(S) A(S) --■A(T)) A(T)) = = r. d(0, A(S) ■-- A(T)) Consequently, r = 2s + 1. By Lemma 5.29 n 2 s +i(S - T) = 2n - 22s and by Lemma 5.30 n2s+1{A{S) - A(T)) = 22s+l - 22s. Since ^ ( 5 - T) = A{S) - A{T), we have n2s+1(S - T) = n2s+i{A(S) - A(T)). Therefore n = 2s + 1 is odd. Hence, if n is even, then the case when only one of S — T and A{S) — A(T) is alternate cannot occur. Consequently, by the above proof A preserves the arithmetic distance. There remains to consider the case when n is odd. Let n = 25 + 1. By the above proof both A\ and Ai1 carry alternate matrices of rank < 25 into alternate matrices of the same rank, and they also carry non-alternate matrices of rank < 25 into non-alternate matrices of the same rank. We shall prove further that both A\ and A^1 carry alternate matrices of rank 25 into alternate matrices of rank 25, then it will follow immediately that they also carry non-alternate matrices of rank 25 + 1 into non-alternate matrices of rank 25 + 1. Thus the case when only one of S — T and ^4(5') — A(T) is alternate cannot occur. So, A preserves the arithmetic distance. We have assumed n ^ 3, then s > 1. Assume that Ai carries an alternate matrix S of rank 25 to a non-alternate matrix Ai(S) of rank 25 + 1. Without loss of generality we can assume that Ai(S) = I^n\ Let

K =

(0 1 1 0

V

\

o /

which is an alternate matrix of rank 2, and 2 < 25. Thus AYl(K) is also an alternate matrix of rank 2. Clearly, K — 1^ is a non-alternate matrix of rank n - 1 - 25. Thus d(K, J(n>) = 25, and then d ^ W , A^(I^) = 2s. But both AiX(K) and > l ^ 1 ( / ^ ) = S are alternate matrices, so AiX(K) —

270 4 - 1 (/(*)) i s

Chapter 5. Geometry of Symmetric an

Matrices

alternate matrix. By Proposition 5.5, d(AiX(K),

must be odd, this is a contradiction.

A^il^)) □

Now let us come to the proof of Theorem 5.4 for the case F = F 2 and n ^ 3. Proof of T h e o r e m 5.4 (the Case F = F 2 and n / 3): We proceed in steps. (i) Let A : Sn(¥2) —> Sn(¥2) be a bijective map and assume that both A and A~* preserve the adjacency. By Lemma 5.31 A preserves the arithmetic distance. In order to include the case n = 3 for latter purpose, we assume that A preserves the arithmetic distance when n = 3. Then we can define maximal sets of rank 2 in <Sn(F2) by Definition 5.4 and prove Propositions 5.13, 5.14, 5.17, 5.18, 5.23, 5.24, 5.25 and Corollaries 5.15, 5.16 as in Sections 5.3 and 5.5. A4i and A42 are two maximal sets of rank 1 in <Sn(F2) whose intersection is {0}, so A(M\) and A(M2) are two maximal sets of rank 1 in 5 n (F 2 ) with a nonempty intersection. By Proposition 5.12, after subjecting A to a bijective map of the form (5.1) we can assume that A(M1)

= M1,

(5.49)

A(M2)

= M2.

(5.50)

Since {0} = Mi fl M2-> we have .4(0) = 0.

(5.51)

It follows from (5.49), (5.50), and (5.51) that A(En)

= E1U

(5.52)

A(E22) = E22.

(5.53)

A{Ct) = A ,

(5.54)

A(C2) = £2.

(5.55)

By Corollary 5.15

(ii) Since d (1 £2 = {0, E12 + En},

(5.51), (5.54) and (5.55) imply

A(E12 + En) = Evt + £ 2 i-

(5.56)

5.7. Proof of the Fundamental Theorem

(III)

271

By Propositions 5.24, 5.25 and (5.54) we may let /

A(En + E12

1

x12 + E2i) = I *i3 \xm

#12

^13

Xln\

0 0

0 0

0 0

0

0

•••

0 )

Since £ n + £ i 2 + £ 2 i and £22 are adjacent, we must have x\z =

• = x\n = 0.

Therefore A(Eu + E12 + E2i) = Eu + £12 + E21. Similarly, A(Ei2 + E21 + E22) — E\2 + ^21 + -#22Let + E22) = {Xij)l xij = xji-

A(En

Since Eu + E22 is adjacent with both En and £22, both /xn-l

\

x12

•••

a?i n \

£12

#22

#2n

X\n

X2n

x

^xn and

X\2

X12

x

X22 ~ 1

x

\X\n

nn '

are of rank 1, from which we deduce x^ = 0

X2n

ln\

2n

&nn '

for 1 < i < n and 3 < j < n,

and xu = #22- But we have (5.56), so xu = ^22 = 1 and hence £12 = 0. Therefore A(En + E22) = En + £22Similarly, from the adjacency of £ n + £ ^ + £21 + £22 with 0, £11+^12 + ^21, and E12 + E2i + £225 we deduce .A(i?ll + £12 + -^21 + E22) = £ l l + E\2 + E21 + £?22le that We conclude A (^^

0(n_2))

= (*'"

0(n_2))

for all X& e S2(¥2).

(5.57)

(iii) We prove by induction on r that after subjecting A to a bijective map of the form (5.1) we can have A X(T)

(

Q(n-r)) = (XiT)

0(»-r))

for all A™ € S r (F 2 ).

(5.58)

Chapter 5. Geometry of Symmetric

272

Matrices

From (5.51) and (5.52) we know that this is true for r = 1. The case r = 2 has been proved in Step (ii). Now we write down the proof of the case r = 3 and leave the general case to the reader. Consider the image of Eiz + £31 under A. By (5.54) we can assume that /

A{E13 + E31)

0

£12

£13

•••

x12

0

0

•••

0

X13

0

0

•••

0

0

0

..

\xln

xin\

0 /

By (5.57) we must have (#i 3 , • • •, xin) ^ (0, • • , 0). There exists a P G GL n _ 2 (F 2 ) such that (&13, •••, xln)P

= (1, 0, •• 0).

Thus after subjecting A to the bijective map

r ,)*r P).

X we have

A(E13

/ 0 x12 1 + ES1) = 0

x12 0 0 0

1 0 0 0 0 0 0 0

\ 0 0 0 Then after subjecting A to the bijective map '/I

\ xu

\

0/

0

/l 1

1

Jf

0\ 0 0 0

1

X12 j(n-3)

'/

\

1

j(n-3) J

we have A(E13 + E31) = E13 + E31.

(5.59)

Notice that we still have (5.57). By Proposition 5.23, there are exactly three maximal sets of rank 2 containing 0 and E13 + E31 in common and they are £ i , £ 3 and £31(1). By (5.54) A{d) = Cu hence A(£3) = £3 or £ 3 1 (1).

5.7. Proof of the Fundamental Theorem (III)

273

If ^ ( £ 3 ) = £31(1), then after subjecting A to the bijective map of <Sn(F2) ^

X

\I + E31)X(I

+ E31),

we can assume that A(C3) = C3. Notice that (5.57) and (5.59) still hold. Since A(0) = 0, we have A(M3) M3. Therefore

=

A(E33) = £33. As in Step (ii) we can prove that 0

0 JfW

JfW Q(n-3)

for all X™

eS2{¥2)

Q(n-3)

and /Xn

0

X13

0

0

0

#13

0

£33

\

/in 0 S13

Q(n-3) y

V

0 0

xi3 0

0

x33

Q(n-3) J

V

for all x n , X13, £33 G F 2 . As in Step (iv) of the proof of Theorem 5.4 for the case of characteristic not two in Section 5.4, from the adjacency of -E11 + E22 + E33 with E n + E22, En + JE 33 , and E22 + E33, we may deduce A(Eu + E22 + E33) = En + E22 + E33, but the details will be omitted. Then as in Steps (v) - (xi) of that proof we can prove A Xi3)

{

0<»-3>)

=

(*(3)

0<"-3>)

f°rallX(3>e«S3(F2),

but the details will also be omitted. (iv) In (5.58) let r = n, then we obtain A(X) = X

for all X G Sn(¥2).

□

Actually we have also proved Theorem 5.4 for the case F = F 2 and n = 2, and the following proposition.

Chapter 5. Geometry of Symmetric

274

Matrices

Proposition 5.32: Let F = F 2 and n = 3. Assume that A : £ 3 ^ 2 ) is a bijective map and preserves the arithmetic distance, that A is of the form (5.2). □ There remains to prove Theorem 5.4 for the case F = ¥2 and n = 3. We begin with a simple lemma on 3 x 3 symmetric matrices over F2. L e m m a 5.33: Let

(

£13'

#11

#12

£12

£22

1

#13

1

Z33 J

be a 3 x 3 symmetric matrix over F2. If S is of rank 1, then xn = £12 = £13 a n d #22 = £33 = 1-

Proof: Since rank 5 = 1, we must have

rankf*22

M = 1.

It follows that #22 = #33 = 1. Since F = F2 and rank 5 = 1, the three rows of S must be the same. Therefore i n = #12 = £13. □ From Lemma 5.33 we deduce L e m m a 5.34: The map (5.3) xu

^12

£13

£11

3i2

X13 \

£12

^22

0

&13

0

£33

Z12 #13

^22 0

0 X33)

xn

^12

£13

£12

#22

1

^13

1

£33

«11 + 1 312 + 1 a?i3 + 1

^12 + 1 ^22 1

*13 + 1

1 #33

for all £ n , £ 12 , #i 3 , £22, £33 £ F 2 is a bijection from <S3(F2) to itself, is equal to its inverse, and preserves the adjacency. Proof: It is enough to prove that if / xu #12 \ Z13

x12

x13'

#22

1

1

£33 ,

2/ll

and

2/12

I 2/12 t/22 > 2/13 0

2/13

0 y33

5.7. Proof of the Fundamental Theorem

(III)

275

are adjacent, i.e.,

rank

f xn - 2/u

#12 -

2/12

3?12 ~ 2/12

3?22 -

2/22

Zl3 -

1

\ ^ 1 3 ~ 2/13

2/13 \

= 1,

1

2>33 — 2/33 /

then so are

(xn

+l

Z12 +

^13 + 1 \ 1

1

U12 + 1

#22+ 1

W +i

1

Z33

/2/n

2/12

2/12

2/22

and

/

2/13 \

0

0

\2/i3

,

2/33/

i.e.,

(

^ Zul l ~- yyni l ++ 1

rank

Zl2 Z12 - 2 /T/i2 1 2 + 1I

X\2 - 2 / 1 2 + 1I Z12

X ^ 1133 ~ ~ t/13 2/13 + + 1 1 ^\

11

£^ 22 2 - 2/22

= 1. = 1-

1

1

^33 -

#13 - 2 / 1 3 + 1

1

£33 - y33

\ ^13-2/13 +

2/33

1

I

But this is clear by Lemma 5.33.

□

Denote the map (5.3) by \Pi. Clearly, \I>i does not preserve the arithmetic distance, for example, **i(0) i ( 0 ) == 0,0,

0

/l

\

**i! I

1

1\

0 1 = 1 0 1 i1 0) o) \ i 1
V

7

1 (° ad

0,

0

0,

1 0 1 : ) ) -

; ) ) -

and ad

1

Similarly, the bijective map \I>2 • /an

Z12

0 \ 1

1^ £ 1 1

£12

#22

3?23

\o fxn

^23

X33J

^12

^12

^22

1 \ x2s '—> X33J

V 1

^23

—> \ f

£12 ^22

3? 2 3

0

£23

^33/

X\l

1

"\

\

z12

^12 + 1

^

0

^12 +

1

1

^22 +

1

*23 + 1

^23 + 1

3^33

\ /

J

Chapter 5. Geometry of Symmetric

276

Matrices

for all i n , i X 2 , ^22, ^23, #33 € F 2 , and the bijective map #3 : in

0

I13

In

0

I13

0

£22

x

23

0

X22

#23

, #13

#23

^33

, £13

#23

^33,

#11

1

#11 1 1 ^22 ^13 + 1 ^23 + 1

^13\

1

£22

^23

, #13

#23

^33 /

#13 + 1 #23 + 1 £33 + 1

for all i n , #i3> #22, ^23, £33 € F 2 , both preserve the adjacency. Denote '0

1

P12 = I 1

0

and

P13

1 and define A12

: X

A13:X

'A2^Pi2 for all I G 5 3 ( F 2 ) , t

PisXP13

for all X

eS3(¥2).

They are bijective maps from <Ss(F2) to itself of the form (5.1). It is easy to verify that * 2 = *4l2#1^12

and #3 =

A13ViA13.

Now we are ready to prove Theorem 5.4 for the case F = F 2 and n = 3. Proof of Theorem 5.4 (the Case F = F 2 And n = 3): Let A : <S3(F2) -* <S3(F2) be a bijective map and assume that both A and A'1 preserve the adjacency. As in the preceding cases, after subjecting A to a bijective map of the form (5.1), we can assume that -4(0) = 0. Since matrices of rank 1 are all the points in 5s(F 2 ) which are adjacent to 0, A carries matrices of rank 1 to matrices of rank 1. Since non-alternate matrices of rank 2 are all the points W in 3(F2) for which d(W/, 0) = 2, A carries non-alternate matrices of rank 2 to non-alternate matrices of rank 2. There are seven 3 x 3 nonzero alternate matrices and twenty-eight 3 x 3

5.7. Proof of the Fundamental Theorem

277

(III)

nonsingular symmetric matrices over F2, (cf. Corollary 4.10 of Wan 1993). Thus there is a nonsingular symmetric matrix in S3(W2) which is carried to a nonsingular symmetric matrix by A. After subjecting A to bijective maps of the form X 1—> *PXP, where P G GL3(W2), we can assume further that A(I) == I. A{I)-J. By Lemma 5.30, there are 3 points W G S3(¥2) for which d(0, W) = 1 and d(W, I) = 2, and they are E n , £12, and E33. After subjecting A to a bijective map of the form X *-* lQXQ, where Q is a permutation matrix, we can assume further that A(EU) === E A(Eu) A(E222)2 ) === E A(E3333) ) === ^E3 33 # 2222, , and -4(£ ♦4(£ 1U -£"11? 3 -. It can be easily deduced from Lemma 5.30 that there are 3 points W G £ 3 ^ 2 ) for which d(0, W) = 2 and d(W, I) = 1, and they are E^ + E22, E1X + E33, and E22 + E33. It follows that

A(E En ++ E22, A(Eun ++ E £22) =: En #22, 22) = A(EuU + A(E En + E33, E33, 33) = + E £33) =■■En and

A(E A(E2222 ++ E + E33. E33. £33) =■EE22 33) = 22 + Consider the image of En + £"12 + E2i under A. Since d(En + E12 + E2i, E22) = 1 and A is bijective, there are four possibilities:

1 I All

l 0

V

\ l = (i l

oj

l

\

0

V

0

,

(l Vi

1

l

\

V 1

(° V

\ 1 , or

0 1 1

(1 1 1\ 1 0 1 .

Vi 1 1/

We distinguish these four cases.

w „(!

i o )-(! i j .

Then we can prove that

A(Xm

_ (X™ 0)

0)

forallX<2)<E<S2(F2).

Chapter 5. Geometry of Symmetric Matrices

278

Consider the image of En + Ei3 + E3\ under A. Since d(Eu + E\z + Esi, E33) = 1, d(En + E13 + E31, En + E12 + En) = 3, and A is bijective, there are three possibilities: A\

(l

l

0

Vi

\

o

(l

l

\

(° ,

= \ o Vi oj V

1

\ , or

M

i o

(1 1 1\ 1 1 1 . \l 1 0

We distinguish further these three cases. (I 1\ (a.l) A\ 0 0. < \1 We can prove that

/l

1'

■u-

/#n

•

o \^13

■

f Xu

£i3\

A\

;

Xi 3 \

.. ^33/

(Xn

for all

. \ ^13

\Zl3

X33 /

2) 6 *(F.).

Then consider the image of E22 + E23 + E32 under A. Since d(E22 + £"23 + ^32? £33) = 1? d(j^22 + E2s + ^32? ^ n + ^12 + E2i + E22) = 1, and A is bijective, there are two possibilities: '0

\ 1

=

M

1 0

(°

1

I

1 0/

\ or

M

(I 1 1 1 1 1

Vi

i o

We distinguish further these two cases. A\ (a.1.1) .4

r

\

0

1 00//

V

11 11 = =1

V I

\ 11 11 .

i1 0o .

We can prove that '0

A

{

xw)-{

xw

forallX (2 >€«S 2 (F 2 )

and then prove in succession that (0 1 1\ /0 1 0\ / 0 0 1\ (0 1 1 \ 1 0 0 , 1 0 1 , 0 0 1 , and 1 0 1

Vi o o/

Vo i o/

Vi

i o

Vi i o j

5.7. Proof of the Fundamental Theorem

(III)

279

are left fixed by A. Therefore all the 7 nonzero alternate matrices are left fixed by A. It follows that A preserves the arithmetic distance. By Propo­ sition 5.32 A is of the form (5.2).

(a.1.2) A\ A\

/0 (°

I

\ 1111

\=

/ Il 11 V1\ 1 1 11 11 .

1

/

ll

1

/

After subjecting A to the bijective map $ i , we can assume that

A\

f°'0

1 1

\ = (°

1 1 ,

V

1

/

1

V

\ /

which reduces to the previous case (a. 1.1). 1

(a.2) A\

0 \l

/0

M

f° 1 1\ . 'I 1 /

O)

After subjecting A to the bijective map ^2^1? we can assume that

ri

M

j A

1

0

U

=

oj

f 1■

J 0

\i

\

.

0

Notice that ^ 2 ^ 1 leaves

(i

1 0 1

V

\ 0

fixed, therefore this reduces to Case (a.l). / 1 1 1\ r (I = 1 1 1 = 1 1 1 . \l \1 00) / \1 Vi 11 00/ y After subjecting A to the bijective map $2>we can assume that A\ (a.3) A\

(/ l1

il\

0

\

1

Af Vi

M = Z1

° oj=

This reduces also to Case (a.l).

1 x l Z" \ ( (b) ,.44 | 11 00 )== (

V

V

Vi

l

11

\J

1

Vi

0

M. oj

Chapter 5. Geometry of Symmetric

280

Matrices

After subjecting A to the bijective map $3*2, we can assume that j1 1

^1

\

1 0

I

1

=

0/

v

l

1 0

\

o;

.

This reduces to Case (a). jfl

(c)

*l

1 0

\

—

(°

\

1 1

o

i-

After subjecting A to the bijective map \P3\P1, we can assume that A

I11 V

1 0

] oj

=

I11

1 1

V

\. i/

But d{En + Ei2 + En, E22) = 1 and d(En + E12 + E21 + E22 + £33, E22) = 3, therefore Case (c) is impossible. (d)

1 Z .4 1

V

x 0

^=

/(iI I1 1' i\ 1i 1i 1 i .

oj

[l

1 0/

After subjecting A to the bijective map $2? we can assume that jri

*

i

1 0

^

\

oj

=

(l

l

1 1

V

\

.

i/

As Case (c), this is impossible.

□

Without essential difficulty the proof of Theorem 5.4 for the case when F is of characteristic two in Sections 5.6 and 5.7 can be generalized to prove the following theorem. Theorem 5.35: Let F and F' be fields of characteristic two, n and n' be integers > 2, and A : Sn(F) —► Sn>(F') be a bijective map. Assume that both A and A'1 preserve the adjacency. Then n = n', F and F' are isomorphic, and A is of the form (5.35) A(X) = aiPX°P

+ S0i

where a e F'*, P e GLn(F'), a is an isomorphism from F to F\ and So € Sn(F'), unless n = 3 and F = F 2 . When n = 3 and F = F 2 , we also

5.8. Application to Algebra

281

have n' = 3 and F' = F 2 , and A is a product of bijective maps of the form (5.2) or of the form (5.3). Conversely, if F is isomorphic to F ' , then any map of the form (5.35) from Sn(F) to Sn(Ff) is bijective, and both the map and its inverse preserve the adjacency. □ Moreover, using the techniques used in the proof of Theorem 5.4 it is easy to prove the following theorem. Theorem 5.36: Let F be a field of characteristic not two, F' be a field of characteristic two, n and n' be integers > 2. Then there does not exist any bijective map from Sn{F) to Sni(F') such that both the map and its inverse preserve the adjacency. □

5.8

Application to Algebra

In this section we assume that both F and F' are fields of characteristic not two. It is known that the set of n x n symmetric matrices over F, Sn(F), form a Jordan ring with respect to the addition (A, B) v—> A + B and the symmetrized

multiplication (A, B)^A.B=l-(AB

Denote this Jordan ring by J(Sn(F)). A : J(Sn(F))

+ BA). A bijective map —*

J(Sn.(F'))

is called an isomorphism of Jordan rings, if A{A + B) = A{A) + A{B)

(5.60)

and A(A-B)

= A(A)-A(B).

As an application of Theorem 5.22 we have

(5.61)

282

Chapter 5. Geometry of Symmetric

Matrices

Theorem 5.37: Let F , F' be fields of characteristic not two and n, n' be integers > 2. If there is an isomorphism of Jordan rings A : J(Sn(F))

—

J(Sn.(F%

then n = n7, F is isomorphic to F 7 , and A is of the form A(X) = aiPX'P,

(5.62)

where a G F 7 *, F is an n x n matrix over F 7 satisfying *PP = a" 1 /, and a is an isomorphism from F to F 7 . Conversely, the map (5.62) from J(Sn(F)) to J(Sn(F')) is an isomorphism of Jordan rings. Proof: The second statement of the theorem is obvious, so we prove only the first statement. We proceed in steps. (i) Let A be an isomorphism of Jordan rings from J(Sn(F)) For A = B (5.60) yields 4 ( 2 4 ) = 2.4(A).

to J(<S n /(F 7 )). (5.63)

Substituting A = 0 into (5.60), since F' is of characteristic not two, we get 4(0) = 0.

(5.64)

By the definition of symmetrized multiplication, we have A • A = A2.

(5.65)

For A = B (5.61) yields 4 ( 4 • 4 ) = 4 ( 4 ) • 4 ( 4 ) . By (5.65) we obtain 4(42) = 4(4)2.

(5.66)

It is easy to verify that ABA = 2 ( 4 • B) • 4 — ( 4 • 4 ) • 5 , from which we deduce A(ABA)

= 4(4)4(5)4(4).

(5.67)

Substituting 4 = I into (5.67), we obtain A{B) = 4 ( 7 ) 4 ( 5 ) 4 ( 7 ) for all B e J(Sn(F)). Since A is bijective, A(I) is nonsingular. Substituting A — I into (5.66) and multiplying both sides by 4 ( / ) _ 1 , we obtain 4 ( 7 ) = 7.

(5.68)

5.8. Application to Algebra

283

Therefore A satisfies (5.63), (5.64), (5.66), (5.67), and (5.68). (ii) An n x n nonzero matrix A over F satisfying A2 = A is called an idempotent. By (5.65) A G J(Sn(F)) is an idempotent if and only if A • A = A. Hence A carries idempotents of J(Sn(F)) into idempotents of J(Sn>(F')). Two n x n matrices A and B over F are said to be orthogonal, if A S = BA = 0. Let A and B be two orthogonal idempotents of J(Sn(F))\ i.e., A 2 = A ^ 0, P 2 = B ^ 0, and AB = BA = 0. By (5.63) and (5.64) 0 = ,4(0) = A(AB + BA) = ^ ( 2 A • P ) = 2A{A) • ,4(J?) = A{A)A(B) Thus ^ ( A ) 4 ( B ) = -A{B)A(A). 0 = A(ABA)

+

A(B)A(A).

By (5.67) we have

= A(A)A{B)A{A)

= -A{A)2A(B)

=

-A(A)A{B).

Therefore A(A)A(B) = A(B)A(A) = 0. Hence ^4 carries orthogonal idem­ potents of J(Sn(F)) into orthogonal idempotents of J(Sn'(F')). An idempotent A of J(Sn(F)) is called primitive, if A = P + (7, where B , C G J(Sn(F)), B2 = B,C2 = C, and BC = CB = 0, imply 5 = 0 or (7 = 0. Clearly *4 carries primitive idempotents of J(Sn{F)) into primitive idempotents of J(Sni(F')). Moreover, we have L e m m a 5.38: Primitive idempotents of J(Sn(F))

are of rank 1.

Proof: Let A be a primitive idempotent of J(<S n (F)), and assume that A is of rank r. Then r > 0 and by Corollary 1.33 there is a P G GLn(F) such that i

PAP=[s1,

5 2 , . . . , f l r , 0, - . - , 0 ]

where Si, 5 2 , • • •, «sr 7^ 0. Let Pi be the submatrix of P " 1 formed by its first r rows, then A=

*Pi[3i, 5 2 , • • - , s r ] P i .

(5.69)

Since A 2 = A, we have *Pl|>l, ^2, ' * ' , 5 r ]Pi *Pi[si, 3 2 , ' ' ' , S r ]Pi = 'Pi[si, 5 2 , ' ' ' , 3 r ]Pi-

Chapter 5. Geometry of Symmetric

284

Matrices

But Pi is of rank r, thus t P P11^[si\s^,-P11*P = [S?, 3 2 - 1 , --- .- T , s ;1 1]]-.

It follows that the rows of Pi are mutually orthogonal. Let vi, v2, • • •, vr be the r rows of Pi, then vt- *ut- = s,"1 for 1 < z < r. By (5.69) we have t A= ■ -\= si *t?iui + s522 v22vv22 + H • ■• h 5 r S^VrVr. *vrur.

It is easy to verify that si *viVi, 5 2 ^2^2, '•> $r tvrvr are r mutually orthog­ onal idempotents of J(Sn(F)). Since A is primitive, we must have r = 1.

□ Conversely, we have L e m m a 5.39: Idempotents of rank 1 of J(Sn(F))

are primitive.

Proof: Let E be an idempotent of rank 1 of J(Sn(F)). Assume that E is not primitive. Let E = B + C, where P , C G J ( 5 n ( P ) ) , P 2 = P ^ 0, C2 = C 7^ 0, and P C = C P = 0, then there is an n x n nonsingular matrix P over F such that

P5P_1 = /(ri)

(

o<—0'

PCP"1 =

J(ra) Q(n-ri-r2) /

where n > 0 and r 2 > 0. Then 1 = rank r a n kPP = rank(P rank(P ++ C) = = r a n k ((PPP5PP~ 1 + PCPP C P1")a ) == nr1 ++ rr22l,

which is a contradiction. (iii) Now we are going to prove that A carries matrices of rank 1 of to matrices of rank 1 of J(Sn'(F')).

□ J(Sn(F))

Let i b e a symmetric matrix of rank 1 over F. Then A can be expressed in the form A = (1*1111, where a e F* and u is a nonzero rc-dimensional row vector over F. There are two possibilities: utu ^ 0 and ulu = 0. Consider first the case utu ^ 0. Let utu = b. Then E = b~l tuu is an idempotent of rank 1 of J(Sn(F)). By Lemma 5.39, E is a primitive idempotent of J(Sn(F)). We have A = EAE. By (5.67) we have A(A) = A(E)A(A)A(E).

5.9. Application to Geometry

285

We know that A carries primitive idempotents of J(Sn(F)) into primitive idempotents of J(S n /(F')). By Lemma 5.38 A(E) is of rank 1. Therefore A(A) is also of rank 1. Then consider the case u tu = 0. There exists an n-dimensional row vector v over F such that

- ( ° V)(in:)crcH/-. a 1

~ ).

0 J

Then (u + v)'(u + v) = 2a" 1 . Therefore Ex = 2~la\u + v)(u + v) is an idempotent of rank 1 of J(Sn(F)), hence is a primitive idempotent. Clearly A = 2AEXA. By (5.63) and (5.67) we have A(A) = 2A{A)A(E1)A{A). As in the above case A(Ei) is an idempotent of rank 1 of J(Sn'(F')). Therefore A(A) is of rank 1. (iv) We have proved that A carries matrices of rank 1 into matrices of rank 1. In the same way we can prove that so does A'1. It follows from (5.60) that both A and A'1 preserve the adjacency. By Theorem 5.22, n = n', F and F' are isomorphic, and A is of the form (5.35) A(X) PXaP + So, A(X) = a%tPX'P S0,

for all X £G J(S J(Snn(F)), (F)),

where a G F'*, P G GLn{F'), a is an isomorphism from F to F1. By (5.64) A(0) = 0, hence S0 = 0. By (5.68) A(I) = I, therefore alPP = I. U

5.9

Application to Geometry

In this section we assume that F is a field of any characteristic and n is an integer > 2. The discussion in the present section is parallel to Section 4.4, so we state the definitions and theorems, and give only the different proofs. Let

/ 0 0 K K =■ ( -\-lM

J<»>\

0 ))>'

which is a 2n x 2n nonsingular alternate matrix in its normal form. A 2n x 2n matrix T over F is called a symplectic matrix with respect to K if t

TKT =--K.

286

Chapter 5. Geometry of Symmetric

Matrices

Clearly 2n x 2ra symplectic matrices with respect to K are nonsingular and they form a group with respect to the matrix multiplication, called the symplectic group of degree 2n with respect to K over F and denoted by Sp2n(F). It is easy to check that the following elements belong to Sp2n(F): 1° 1°

(( AA

2° 2°

(f17

3° 3°

((

tt (( A A

__ 11 }} )) ,, where where A A G GG G ii nn (( P F )) ;;

jj )Y, where where *5 *5 = = 5; 5; rT

7

T

nere 2 ) ? wwhere »/ ), J2 = = «/ J is is aa diagonal diagonal matrix. matrix.

Let m be an integer, 0 < m < ra, and P be an m-dimensional subspace of the 2n-dimensional row vector space F^2n\ Define PPL1- = {v£F(2n>>\vKtP

= Q}

and call P x the dual subspace of the subspace P relative to K. Clearly we have PLK*P = 0 and by Theorem 1.27 d i m P 1 = 2n — m. It is easy to see that the map P*—>PL from the set of subspaces of F( 2n ) to itself is bijective and has the following properties: for any two subspaces P and Q

PCQ^P'DQ 1 , and for any subspace P dimP + dimP1 = = 2n and ( P 1 ) 1 = P. If we regard an m-dimensional subspace P as an (m — l)-flat in PG(2n — 1, P ) , then PL is a (2n - m - l)-flat, called the dual flat of P . Therefore the map P -> P 1 from the set of flats in PG(2n - 1, F) to itself is a polarity of PG(2n - 1, P ) , called the symplectic polarity defined by the 2n x 2n nonsingular alternate matrix K, or simply the symplectic polarity.

5.9. Application to Geometry

287

If PL — P , the P is called self-dual and the corresponding subspace P is also called self-dual. Clearly self-dual subspaces are of dimension n and /j(n)

Q(TI)\

is a self-dual subspace. Parallel to Proposition 4.18 we have P r o p o s i t i o n 5.40: A subspace W is self-dual if and only if dim W = n and WK*W = 0.

□ An m-dimensional subspace P is called totally isotropic (with respect to A') if PK *P = 0. A totally isotropic subspace is called maximal totally isotropic if its dimension is equal to the maximum of the dimensions of totally isotropic subspaces. Maximal totally isotropic subspaces are of dimension n. Parallel to Propositions 4.19 and 4.20 we have P r o p o s i t i o n 5.41: A subspace is self-dual if and only if it is a maximal totally isotropic subspace. □ P r o p o s i t i o n 5.42: Let W be an n-dimensional subspace. Write a matrix representation W of the subspace W in the block form (X Y),

(5.70)

where both X and Y are n x n matrices. Then W is self-dual if and only if XlY

is symmetric, i.e., XtY

= YtX.

(5.71) D

Let W be a self-dual subspace. If in the block form (5.70) of W rank X = n, then W has a matrix representation (J ( n ) X-XY).

(5.72)

From (5.71) we know that X~*Y is an n x n symmetric matrix. Hence the set consisting of all self-dual subspaces is called the projective space ofnxn symmetric matrices over F and is denoted by VSn(F). It is also called the

288

Chapter 5. Geometry of Symmetric

Matrices

dual polar space of type Cn. The self-dual subspaces are called the points of VSn(F). Any matrix representation (5.70) of a point W G VSn(F) is called a homogeneous coordinate of the point. If the matrix X in (5.70) is of rank < n, then the point W is called a point at infinity. If the X in (5.70) is of rank n, the W is called a finite point, and it has (5.72) as one of its homogeneous coordinates and the n x n symmetric matrix X~XY as its non-homogeneous coordinate. As in Section 4.4, restricting the actions of GL2n{F) on F^2n^ and on the set of subspaces of F^ to Sp2n(F), we obtain actions of Sp2n(F) on F<2n) and on the set of subspaces of F^2n\ Parallel to Propositions 4.21 - 4.26, Corollary 4.27, and Definitions 4.6 4.7, we have Proposition 5.43: Sp2n(F)

acts transitively on VSn(F).

□

Definition 5.7: Two points W\ and W2 of VSn(F) are said to be of arith­ metic distance r, denoted by ad(Wi, W2) = r, if dim(Wri U W2) = n + r. When r = 1, they are said to be adjacent □ Proposition 5.44: Let W\ and W2 be any two points of VSn(F). ad(Wi, W2) =

Then

r<mkW1KtW2.

□ Proposition 5.45: Let Wu W2 and W3 be any three points of Then

VSn(F).

1° ad(Wi, W2) > 0; ad(Wi, W2) = 0 if and only if W1 = W2. 2° ad(Wi, W2) = ad(W 2 , Wk). 3° ad(Wi, W2) + ad(W 2 , W3) > ad(Wk, W 3 ).

D

Proposition 5.46: Let Wi and W2 be two points of arithmetic distance r in VSn(F). Then they can be carried simultaneously to

(7 0) and

^

Q

/(n _ r)

Q

0(n.r)J.

D

5.9. Application to Geometry

289

P r o p o s i t i o n 5.47: Sp2n{F) leaves the arithmetic distance between any two points of VSn(F) invariant. Moreover, for any fixed r, 0 < r < n, the pairs of points of VSn(F), which are of arithmetic distance r, form an orbit of SP2n{F). □ Notice an important distinction between Sn(F) and VSn(F). For any fixed r, 0 < r < n, the pairs of points of Sn(F), which are of arithmetic distance r, do not necessarily form an orbit under the group of motions GSn(F), however, the pairs of points of VSn{F), which are of arithmetic distance r, do form an orbit under Sp2n(F). Definition 5.8: Let W, W £ VSn{F). When W ^ W, they are said to be of distance r, denoted by d(W, W) = r, if r is the least positive integer for which there is a sequence of r + 1 points Wo = W, W\, W> • • •, Wr = W such that W{ and Wi+i are adjacent, i = 0, 1, 2, • • •, r — 1. When W = W', define d(W, W) = 0. □ P r o p o s i t i o n 5.48: For any two points W, W £

VSn(F)

ad(W, W ) = d(W, W 7 ). D

C o r o l l a r y 5.49: Let >1 be a bijective map from VSn(F) to itself and assume that both A and A'1 preserve the adjacency of pairs of points VSn(F), then A preserves also the arithmetic distance between any pairs of points in VSn(F). □ We call a 2n x 2n matrix T over F to be generalized symplectic with respect

to K\i l

TKT = aK,

where a £ F*. The set of all 2n x 2n generalized symplectic matrices with respect to K over F also forms a group with respect to the matrix multipli­ cation, called the generalized symplectic group of degree 2n with respect to K over F and denoted by GSp2n(F). Every element T £ GSp2n(F) defines a bijective transformation on VSn(F) W>—+WT

Chapter 5. Geometry of Symmetric

290

Matrices

which preserves the arithmetic distance between any pair of points. In par­ ticular, both the map and its inverse preserve the adjacency of pairs of points of VSn{F). Conversely, we have the fundamental theorem of the projective geometry of symmetric matrices, which reads as follows. Theorem 5.50: Let n > 3 and F be a field of any characteristic. Let A be a bijective map of VSn{F) to itself and assume that both A and A'1 preserve the adjacency of pairs of points. Then A is of the form W h— ► W°T ^W°T

w*-

forall all WW €e VS for VSn(F), n(F),

(5.73) (5.73)

where a is an automorphism of F and T G GSp2n(F). Conversely, any map of the above form from VSn(F) to itself is bijective and both the map and its inverse map preserve the adjacency of pairs of points of VSn(F). To prove this theorem we need the following preparations. Let W e VSn(F)

and write

W = (wi w22 -' •"• W =(w W l

U>2n), W2n),

where Wi, w2j ■ • •, w2n are the 2n column vectors of W. Let 5 be an integer, 0 < 5 < n, and z'i, z2, • • •, zs, j i , j 2 , - , jn-s be a permutation of 1, 2, • • •, n, 1 < z'i < i2 < • - • < is < rz, 1 < ji < j 2 < • • • < jn-s < n. Define Sili2...is = {W eVSn(F)\(wil

••• wis wn+jl

••• wn+jn_a) is nonsingular}.

When s = 0, we denote

S,= {W{WG € VS S = VSnn(F)\(u (F)\(w>n+l n+1

wn+2' Wn+2

•'•W2n) w2n)is nonsingular}. is nonsingular}.

r* T .o-m YY-» :»0 4.29 A. OQ and a n n A 3 0 we n m hi 3T7A Then-» TM»fs»ll
L e m m a 5.51: For any W G Si^...^, multiplying W from the left by the inverse of (—wtl — W{2 • • • — W{s i^n+ji Wn+j2 ''' wn+jn-3)i w e obtain a matrix representation of W whose z'l-th, 22-th, • • •, z s -th, ( n + j ^ - t h , (n-\-j2)th, • • •, (n + j n _ s )-th columns form the matrix //<•> /

(

»

■

-»)

5.9. Application to Geometry

291

and the remaining n columns can be arranged into an n x n symmetric matrix Z == (w (wn+il n+il

wn+i2 ■■ wn+i , n+i2 ■••• n+is

w whh

w wkj2 ■••• ■•

wjn_a).

* > ) * - . ) •

(5.74)

Moreover, the map

sn (F) w >—+ zZ W

<^'l<2 —is

— ►

,

►

is a bijective which preserves the arithmetic distance of any pair of points.

a The n x n symmetric matrix Z in (5.74) is called the coordinate of the points W G <Stlt-2...»v

non-homogeneous

L e m m a 5.52:

u

= u

■PS (F) = U 'PSnn{F)=

U

$.*••■.■.. ^ t l t ' 2 — is'

0<s
D

Parallel to Lemma 4.32 we have L e m m a 5.53: Let n > 2, A be a bijective map from Sn(F) to itself, and assume that both A and A'1 preserve the adjacency of pairs of points of Sn{F). Let io be a fixed index, where 1 < i0 < n. Assume that A leaves every point of Sn(F) whose (z0, io)-entry is nonzero fixed, then A leaves every point of Sn(F) fixed unless n = 2 and F = F 2 . Proof: We consider only the case z0 = 1, since the other cases can be treated in a similar way. Assume that A leaves every point whose (1, 1)entry is nonzero fixed. Let

(°

X #22 22

•• •' • xiXin n\ \ X x2n *'" ' 2n

\Xin \Xln

XX2n2n

•

/ 0

xX ==

x 12

X12

Xnn '

By hypothesis we can assume that

A(X) =

/1 0

x ^12

"

X

^22

' '

U

Ui»

x

*

2n

'

'

'

x

m

}

X

2n

X

nn)

•

292

Chapter 5. Geometry of Symmetric

Matrices

Let /A Xx =

#12

\ X\n

x12

•••

xln\

#22

••'

#2*

X2n

•••

Xnn

/

where A £ F*. Since X and Xx are adjacent and A(Xx) = -X'A by hypothesis, A(X) and XA are adjacent for all A € F*. That is, / ^12

rank

\Xln

#12

"12 a;;* 22

X

X

ln

#12

"In

#ln

#22

C

#2n

X

X

2n

2n

nn

1n

\

= 1 for all A e F*. (5.75)

X

nn

/

If F 7^ F 2 , then we must have £*■ = # t j for 2 < i < j < n and hence also x\{ = xu for 2 < z < n. Therefore A(X) = X. Now consider the case F = ¥2 and n > 3. If we can prove that #jt- = #itfor 2 < z < n, then we have also x*j = X{j for 2 < i < j < n. Suppose on the contrary that x^ ^ xu for some i, where 2 < i < n. Without loss of generality we can assume that i — 2. Then x^ — #12 == 1- Since we have only A = 1 G F£, we have also x^2 — #22 = 1. Thus (5.75) becomes / rank

1

1

1

1

#13 ^13 X23 ~ # 2 3

^ 1 3 ~~ ^ 1 3

^ 2 3 ~~ ^ 2 3

# 3 3 ~" ^ 3 3

\ xln

X

x

ln

2n

~~ X2n

x 3n

X

ln

~~ Xln

°2n

#2n

'In'

#3n

X

~~ # 3 n

nn

x

nn

\

= 1.

(5.76)

/

Let 1

#12

#13 + 1

#14

#ln\

#12

#22

#23

#24

#2n

#13 + 1

#23

#33 + 1

#34

#3n

#14

#24

#34

#44

#4n

V #ln

#2n

#3n

x

/

X\3 =

An

Clearly X and X13 are adjacent. By hypothesis, A(X13)

= XX3.

Therefore

5.9. Application to Geometry A(X)

and .X13 are adjacent, i.e., the rank of

/

1 1

1 1 X

23

* ^14

\

293

^ln

^14

x

ln

—

2n

1

Xj4 -

#34

^34

x

X

1n

^ T l

'

Xir

^24

*

# 2 n ~"

#34

'

x

~

X

*44

x44

•

x

~

X

X

3n

''In -

34

L

^ 3 3 ~~ ^ 3 3 ~" 1

X14

X

^23 ~~ *^23 ^23

^24 ~~ ^24

x

Xl3 -

^13

4n

x

4n

3n 4n

\

2n

3n 4n

x

nn

/

(5.77) is equal to 1. From (5.76) we have ^13

#13

^23 - ^ 2 3

= 0.

From (5.77) we have u

13

*13 ~

1

^23 ~~ • r 23

= 0.

This is a contradiction. Therefore £ j 2 = ^12•

^

The case F = ¥2 and n = 2 is an exceptional case of Lemma 5.53. In fact, the map

n o w i o\ n i\ vo 0; vo oy' vi oy

/1 i\ u or

U l ) - * lo 1)' vi i)~* u i ) '

«!!)-(! IMS !)-(?!)•

« i n s ?)•«!)-«!:) satisfies the hypothesis of that Lemma, but does not satisfy the conclusion. Moreover, it also preserves the arithmetic distance between any pair of points of 5 2 (F 2 ). Proof of Theorem 5.50: We proceed in steps. (i) Let A be a bijective map from VSn(F) to itself and assume that both A and A"1 preserve the adjacency of pairs of points. By Corollary 5.49 A preserves also the arithmetic distance of pairs of points. As in Step (i) of the proof of Theorem 4.28, after subjecting A to a symplectic transformation, we can assume that A leaves every point of Sfixedand A(I 0) = (/ 0).

Chapter 5. Geometry of Symmetric

294

Matrices

(ii) Let ii, z2, • • •, is, j i , 32, ■ • •, jn-s be a permutation of 1, 2, • • •, n such that 1 < h < i2 < •" < is < n and 1 < j i < j 2 < • • • < jn-s < n. Denote by Wi1i2...ia the point of VSn(F), which has a matrix representation whose z'l-th, z 2 -th, •••, z s -th, ji-th, j 2 -th, • • •, j n - s - t h columns are ei, e 2 , • • •, e s , e 5 + i, e 5+2 , * * •> e n , respectively, and all other columns are O's. We assert that A leaves each Wi1i2...is fixed. We prove by induction on s. Clearly, Wj, = (0 / ) , so by (i) our assertion is true for 5 = 0. Now let s > 1 and assume that our assertion is true for 5 — 1. Let us prove that it is also true for s. It is enough to consider the case (z'i, i 2 , • • •, is) — (1, 2, • • •, s).

Let

A(W12...S) = (x

n

where both X and Y are n x n matrices. As in Step (ii) of the proof of Theorem 4.28, by Proposition 5.44 we have rank X = s and rank Y = n — s. Let X = (x1 x2

• • • iB)

and

Y = (t/i y2 • ■ • y n ),

where 3?i, 3?2, • • •, xn are column vectors of X and 2/i, y 2 , * * •, 2/n are those

of r. Consider first the case 5 = 1. Then r a n k X = 1 and r a n k F = n — 1. W\ is adjacent to all the points (aEu

JW) and (a(£ l t - + £ n ) /(»>), a G F\

1 < * < n.

and all these points lie in S+. So (X y ) is also adjacent to all of them. It follows that rank(xi — cty1 x2 • • • xn) = 1, rank(x! - ayt- x2 • • • #,-_! xt- - a ^ z 1 + 1 xn) = 1, 1 < t < n.

(5.78)

From rank(X y ) = n and (5.78) we deduce x2 = x3 = • • • = xn = 0. From rank X = 1 we deduce xi ^ 0. The last n - 1 equations in (5.78) becomes rank(zi - ayt- - T/i) = 1, 1 < i < n. Since n > 3 and rank(X y ) = n, we must have yi = 0 and (xi y2 • • • y n ) is of rank n. Let P = (xt y2 • • • y n ), then (X y ) = (x! 0 . . . 0 y 2 . . .

yn) = P W i .

5.9. Application to Geometry

295

Hence A(W1) = W1. Then consider the case s > 1. Clearly, Wi2...s is adjacent with W\2...s-\, W12...s_2,s, * * *, W23...S' By induction hypothesis each of them is left fixed by A. So (X Y) is adjacent with all of them. Thus rank(-yi

- y2 ••• - ys-i

rank(-yi

rank(xi

xs xs+1

••• xn) = l,

- y2 ••• x s -i

- ys xs+1

••• x„) = l,

- y2 ••• - y s _i

- ys xs+1

••• x n ) = l.

But we also have rank(#i x2

••• xn) = s,

rank(yi y2 ••• yn) = n-s. Consequently, xs_|_i = xs+2 = • • • = xn = 0 and t/i = y2 = * • • = ys = 0. Then P = ( i i a;2 • • • xs ys+i ys+2 • • • yn) is of rank n. Therefore (X Y) = {Xl

••• xs 0 . . . 0 y s + 1 ..- y n ) ^ ^ - 1 ^ ^ . . , .

Hence ^ ( ^ i 2 . . . s ) = Wi2»... (iii) We can prove that A leaves <Stll-2...ta fixed elementwise in the same way as Step (iii) of the proof of Theorem 4.28. (iv) We conclude also in the same way as Step (iv) of the proof of Theorem 4.28 that A leaves VSn(F) fixed elementwise. The proof of Theorem 5.50 is now completed.

□

Without essential difficulty Theorem 5.50 can be generalized as follows. Theorem 5.54: Let F and F1 be fields and n and n' be integers > 3. Let A be a bijective map from VSn{F) to VSnt(F') and assume that both A and A~v preserve the adjacency. Then n = n7, F is isomorphic to F ' , and A is of the form W ^

WaT

for all W G VSn(F),

(5.79)

where a is an isomorphism from F to F' and T G GSp2n(F). Conversely, any map of the form (5.79) from VSn(F) to VSn(F') is bijective and both the map and its inverse map preserve the adjacency of pairs of points. □

Chapter 5. Geometry of Symmetric

296

5.10

Matrices

Application to Graph Theory

At first, let F be a field of any characteristic. Now call the points of Sn(F) vertices and define two vertices 5i and 52 to be adjacent if rank(S'i — S^) = 1. Then we obtain the graph of symmetric matrices. Denote this graph by T(Sn(F)). From Proposition 5.5 we deduce Proposition 5.55: > 2. Then T(Sn(F)) or n is odd, then its then its diameter is

Let F be a field of any characteristic and n be an integer is a connected graph. If F is of characteristic not two diameter is n; if F is of characteristic two and n is even, n + 1. □

Theorems 5.22, 5.35, and 5.36 can be interpreted as follows: Theorem 5.56: Let F and F' be fields and n and n' be integers > 2. Assume that there is a graph isomorphism from the graph of symmetric matrices T(Sn(F)) to T(Sni(F')), then n = n', F is isomorphic to F\ and the graph isomorphism is of the form (5.35) X*-+a

tpX'P

+ So

for all X G Sn(F),

where a G F'*, P G GLn(Ff), a is an isomorphism from F to F', and So e Sn(F'), unless n = 3 and F = F 2 , or n' = 3 and F' = F 2 . When n = 3 and F = F 2 , so n' = 3 and F' = F 2 , and conversely; and in this case, the graph isomorphism is an automorphism and is a product of automorphisms of the form (5.2) or of the form (5.3). Conversely, if F is isomorphic to F\ then any map of the form (5.35) is a graph isomorphism from T(Sn(F)) to T(Sn(F')). When n = 3 and F = F 2 , the map (5.3) is a graph automorphism

ofr(53(F2)).

□

Corollary 5.57: Let F be a field of any characteristic and n be an integer > 2. Then the group of graph automorphisms of the graph T(Sn(F)) consists of the graph automorphisms of the form (5.2) unless n = 3 and F — F 2 . When n = 3 and F — F 2 , the group of graph automorphisms of the graph r(<S3(F2)) is generated by graph automorphisms of the form (5.2) and of the form (5.3). D We may define two vertices Si and 5 2 to be adjacent if rank(Si — 5 2 ) = 1 or 2.

5.10. Application to Graph Theory

297

Then we obtain also a graph. Denote this graph by T*(Sn(F)). Clearly, if F is isomorphic to F\ a map of the form (5.35) from T*(Sn(F)) to T*(Sn(F')) is a graph isomorphism. When n = 2, there are isomorphisms from T*(Sn(F)) to r*(S n (F')) not of the form (5.35). In fact, T*(S2(F)) is a complete graph. Thus if \F\ = \F'\, then any bijective map from T*(<S2(F)) to T*(S2(F')) is a graph isomorphism. For n > 3, we distinguish the cases when both F and F' are of characteristic not two or two. Consider first the characteristic not two case. Theorem 5.58: Let F and F' be fields of characteristic not two, and n and n' be integers > 3. If there is a graph isomorphism from the graph T*(Sn(F)) to the graph T*(Sn*(F% then n = n\ F is isomorphic to F ' , and the graph isomorphism is of the form (5.35). Conversely, if F is isomorphic to F', then the map (5.35) is a graph isomorphism from the graph T*(Sn(F))

to r*(sn(F')). To prove this theorem we need some preparation. For any S G Sn(F), S± = {Te

Sn(F)

define

| rank(T - S) < 2 }.

For any nonempty subset 11 of «Sn(F), define

nL = H sL. sen Clearly, H± = {Se

Sn{F) | rank(5 - T) < 2 for all T G K }.

Consequently,

ncRLL

and Tl CS

implies HL D SL.

L e m m a 5.59: Let n > 3, and 5i, 52 G Sn(F). 1. Then |{ Su S2 } ^ | = \F\.

Assume that rank(S , i-5 2 ) =

Proof: (i) Suppose that Si and S2 are carried into I \ and T 2 , respectively, under the transformation (5.1), then {^i, S2}L is carried into {Ti, T2}L and

Chapter 5. Geometry of Symmetric

298

Matrices

{ft? ft}1J" into {Ti, T2}"1"1". It is clear that Si and ft can be carried under a transformation of the form (5.1) into 0 and aEn with a ^ O , respectively. Therefore we can assume that Si = 0 and ft = aEn, aEn, 0a 5^^ 00..

s2

(ii) Let S G {ft, ft}"1"1. We are going to prove that S = SnEn

for some

s n G F. Write S = (sij)i
It follows that s^ = 0 for 3 < i < j < n and Sij = s 2 j = 0 for 3 < j < n. Similarly, from E13 + E31 and E33 G {£1, S2 } L we deduce s22 = 0 a n d S12 = 0, respectively. Therefore 5 = 5 n £ n . Let <S = {xEu\x G -F}. Then

{ft,

ft}±xcs.

(iii) We prove that if T G {ft, ft}1, then T G 5 1 . Write

T12\ \ Tl2

rp = _ ( hi T =

VT12 ~VTi2

T22J' where *u G F1, Ti 2 is a 1 x (ra — 1) matrix over F , and T22 is an (n — 1) x (ra — 1) symmetric matrix over F. Since T G {ft, ft}1, we have rankT < 2 and rank(T — aEn) < 2, from which follows rankT 2 2 < 1. If rankT 2 2 = 0, then T eCi and clearly T G 5 1 . If rankT 2 2 = 1, then there is a Pi G GL n _i(F) such that M P\TTIP-\ M A = — [b, lk 0, 0, 0, 0, •• •• ••, 0], where be F*. Let

P--■

( ' «

)

■

Since rank T < 2, we have l

Itn

PTP = M12

l

12

6

\

0(»-»)J'

Thus ' P T P 6 5 X . Since ' ( P " 1 ) . ? ^ - 1 ) = S, we have T G <SX. Therefore {Si, S 2 } 1 C S1. Consequently, {Su S2}L± 2 SXL. But S C <SXX. Hence S C {51; S 2 } x x .

5.10.

Application

to Graph Theory

299

(iv) We conclude t h a t {Su Therefore \{SU

S2}LL\

S2}±±

=

{xE11\xeF}.

= \F\.

D

L e m m a 5 . 6 0 : Let n > 3 and S i , S 2 G Sn(F). LL

2. T h e n { S i , S2}

Assume that r a n k ( S i ~ S 2 ) = LL

= {Su

S 2 } and |{Si, S2} \

= 2.

P r o o f : As in L e m m a 5.59 we can assume t h a t Si = 0

and

S2 = aiEu

Let S G { S i , S2}J"~L and write S = (sij)i
a a

+ a2E22i

i2

^ 0.

where Sij = Sji (1 < z, j <

T h e n as in t h e proof of L e m m a 5.59, we can prove t h a t stJ- = 0 for

3 < i < j < n and Sij = S2j = 0 for 3 < j < n. Clearly, mEn

+ XE33,

a2E22 + XE33 G {Si, S 2 } 1 for all X e F.

T h e n we

have s n - ai 5i2

S12 522

$11 Sl2

5i2 ' = 0. ^22 - CL2

T h a t is, SllS22 ~~ 5'12 12 "" a l S 2 2 = 0 0. S11S22 — s\2 — a2su If S11S22 "" 5 i2 su

=

0? t h e n ai«s22 = a 2 S n = 0.

(5.80)

Since a i a 2 7^ 0, we have

= s22 = 0. T h e n from (5.80) we deduce 512 = 0. T h a t is, S = 0. It

remains t o consider t h e case 511622 — s\2 / 0. It is easy to verify t h a t lax X

\ a2 fi

fi a^"1a2"1(A2a2 + fi2a>i)

G {Si, S2}L

for all A, /z G F.

Q(n-S) J

Thus •S11 - Q>\ s12 S\2 5 2 2 - 0>2 —X —fj,

-A = 0 —fl —aj"1a2~1(A2a2 + / / 2 a i )

for all A, /x G F.

E x p a n d i n g this d e t e r m i n a n t , we o b t a i n [ a 7 1 a 2 3 n - 0 ^ ( 5 1 1 5 2 2 - 5 i 2 ) ] A 2 + 23i2A// + [aia2" 1 522-a 2 " 1 ( 5 ii 5 22-5i2)]A i 2 = 0

300

Chapter 5. Geometry of Symmetric

Matrices

for all A, fie F. Thus a^1a2s11 — a^1(s1is22 — s12) = a ^

s22 — a2 (sus22 — s12) = S\2 = 0.

Therefore 5n = ai, s22 = a2. That is, S = aiEu + ^2^22 = S2. {Su S2}^ = {S1, S2}.

Hence □

Proof of Theorem 5.58: Let A be a graph isomorphism from T*(Sn(F)) to T*(Sn*(F')). It follows from Lemmas 5.59 and 5.60 that for Su S2 G Sn(F) rank(5i - S2) = 1 if and only if rank(.4(5 , i) - -4(52)) = 1. That is, both A and A'1 preserve the adjacency. By Theorem 5.22, A is of the form (5.35).□ Corollary 5.61: Let F be a field of characteristic not two and n be an integer > 3. Then the group of graph automorphisms of T*(Sn(F)) consists of the graph automorphisms of the form (5.2). □ Before we come to the characteristic two case, we recall that a field is called perfect, if it is of characteristic 0 or if every element of it is a p-ih power when it is of characteristic p and p ^ 0. In particular, if F is a perfect field of characteristic two, then every element of F is a square. We know that all finite fields are perfect. Now let us consider the graph r*(<S n (F)), where F is a perfect field of characteristic two. T h e o r e m 5.62: Let F be a perfect field of characteristic two and n be an integer > 3. Then the graphs T*(Sn(F)) and T()Cn+i(F)) are isomorphic graphs and the map ICn+1(F))

a K)

—»

Sn(F))

■+ K +

(5.81)

l

aa,

where a is a 1 x n matrix and K is an n x n alternate matrix. Proof: At first we prove that the map (5.81) is a bijection. It is sufficient to prove that it has an inverse map. Let O =

\sij)l
be an n x n symmetric matrix over F , then Sij = Sji for all (i, j).

5.10. Application to Graph Theory

301

Since F is perfect, there are elements a i , a2, •••, an such that a\ = su for 1 < i < n. Clearly, a i , a2> * • * > #n are uniquely determined. Let a = (ai, a2, • * •, « n ), then it is easy to verify that the map Sn(F)) Sn(F))

— --

/C n + 1 (F)) £«+!(*■))

5 H-

*

\*a

S+*aaJ

is the inverse map of (5.81). Now let us prove that the map (5.81) preserves the adjacency. Let

{*a K) (•1 *)

and and

(«°6 (i k)) b

be two adjacent points in /C n +i(.F), i.e., 0 a + 6 \ 2= 2. rank rank (U% ( = ' a + *b M K + J KJ-

(5-82)

We want to show that

#K + l'aaaa

and K *bb Kix + and +*bb

are adjacent, i.e., t taa r<mk(K rank(if + Kt K1++ aa+

t bb) === 11 or 2. + *66) or 2.

We distinguish the following two cases. (a) if + # i = 0. From (5.82) we deduce a + b ^ 0. Then clearly rank(if + # i + laa^ lbb) = rank('(za + *bb) = 1 or 2. (b) # + # ! ^ 0. Then r a n k ( # + Kx) = 2. There is a Px G G L n ( F ) such that / '0 0 11 \ t P1(K + +K K11)P | 1 0 *Pi(ff )P1= = 1 1

\

n2 Q(n-2) o< ->/

By (5.82), we have

\ /

'('

0

a + b \ (I

t P1J\ta+ <
/ 0 Ci * ) = c2

^

d 0 1

c2 1 0

\ ?

o( n _ 3 ))

Chapter 5. Geometry of Symmetric

302

Matrices

where ( C l , c 2 , 0 , ■•-,0) = (a + 6)PiThere is a P 2 € SL2(F)

such that ( d , c 2 )P 2 = (1, 0).

Let

P = Pl 2

\ /(*-*) J '

then

*P(K + # i ) P =

/0 1 1 0

V

0(""2)

and (a + 6)P = ( l , 0 , . . . , 0 ) . Let a P = (ai, a 2 , • • •, a n ), then 6P = (ax + 1, a 2 , • • •, a n ). Thus (

1 l + a2

1 + a2

a3

an\

*/>(# + # ! + *aa + tbb)P =

whose rank is equal to 1 or 2. This proves that the map (5.81) preserves the adjacency. When F is a finite field, the number of pairs of adjacent points is also finite, hence the adjacency preserving of the map (5.81) implies the adjacency preserving of its inverse. When F is not finite, the proof that the inverse map of (5.81) preserves the adjacency is left to the reader. □ Now let F be a field of any characteristic and n be an integer > 2. Similarly, we call the points of VSn(F) the vertices and define two vertices W\ and W2 to be adjacent if W\ U W2 is of dimension n + 1. Then we obtain the graph of the projective space of n x n symmetric matrices over P , which is denoted by T(VSn(F)) and is also called the dual polar graph of type Cn. From Propositions 5.46 and 5.48 we deduce

5.11.

Comments

303

Proposition 5.63: Let F b e a field of any characteristic and n be an integer > 2. Then the graph T(VSn(F)) is connected, distance-transitive, and with diameter n. D Theorem 5.54 can be interpreted as follows: Theorem 5.64: Let F and F' be fields and n and nf be integers > 3. If there is a graph isomorphism from the graph T(VSn(F)) to T(VSn'(F')), then n = n', F is isomorphic to F\ and the graph isomorphism is of the form (5.79) W ■—► W'T, where a is an isomorphism from F to F' and T G GSp2n{F')>

n

Corollary 5.65: Let F be a field of any characteristic and n be an integer > 3. Then the group of graph automorphisms of T(VSn(F)) consists of the graph automorphisms of the following form W i—► WaT, where a is an automorphism of F and T G GSp2n(F).

5.11

Q

Comments

The fundamental theorem of the geometry of symmetric matrices over any field of characteristic not two was proved by L. K. Hua, cf. Hua 1949. He used the method of the construction of involutions. It was then proved by the author by the method of maximal sets (of rank 1 and of rank 2), cf. Wan 1994a. Sections 5.1 - 5.4 follow Wan 1994a. The fundamental theorem of the geometry of symmetric matrices over any field of characteristic two when n > 3 was proved by the author, cf. Wan 1994b and when n = 2 it was proved by J. Gao, Z. Wan, R. Feng, and J. Wang, cf. Gao et al. 1996. Sections 5.5 - 5.7 follow Wan 1994b and the proof of Theorem 5.4 for the case F being of characteristic two and n = 2 in Section 5.6 follows Gao et al. 1996. The results of Section 5.8 are due to the author, cf. Wan 1994a.

304

Chapter 5. Geometry of Symmetric

Matrices

The fundamental theorem of the projective geometry of symmetric matrices (i.e., Theorem 5.50) is due to W. L. Chow, cf. Chow 1949. Its derivation from the fundamental theorem of the geometry of symmetric matrices is due to M. Liu, cf. Liu 1965, but she assumed that the ground field is of characteristic not two, because at that time only the fundamental theorem of the geometry of symmetric matrices over any field of characteristic not two was proved. Section 5.9 follows Liu 1965 with some modification. Most of the results in Section 5.10 are merely translations of some foregoing results into the graph theory language. But the following should be men­ tioned. When F = ¥q is a finite field of characteristic not two, the graph r*(<Sn(Fg)) is defined and proved to be distance-regular by Y. Egawa, cf. Egawa 1985. Theorem 5.58 is due to the author, cf. Wan 1994a. When F = Wq is a finite field of characteristic not two, Corollary 5.61 can be found in Brouwer et al. 1990.

Chapter 6 Geometry of Hermitian Matrices 6.1

The Space of Hermitian Matrices

Let D be a division ring which possesses an involution. Denote the involution of D by —, i.e., - : D —► D a i—► a, which is a bijective map satisfying the following conditions: for all a, b £ D

a + b = a + 6, ab = 6a, and a = a. Let F = {ae D\a = a}, and define the trace map Tr:

D

—+

a

i—► a + a,

F

and the norm map N:

D a

—-> F i—► aa. 305

Chapter 6. Geometry of Hermitian

306

Matrices

We make the following assumptions: Assumption 1° : F is a proper subfield of D and is contained in the center of D. We call F the fixed field of the involution — of D. Assumption 2° : The map Tr is surjective, i.e., any a G F can be expressed as a = a + a for some a G D. Notice that the Assumption 1° excludes the case when D is a field and — is the identity map, thus it is stronger than the Assumption 1° we made in Section 1.5. We are going to study the set of all n x n hermitian matrices over F , which will be called the space of n x n hermitian matrices over D and denoted by Hn(D). The n x n hermitian matrices over D are called the points of the space. With the space Ti^D) we associate naturally a group of motions which consists of transformations of the form Z »-— lTZP

+ H

for all Z G Hn{D),

(6.1)

where P £ GLn(D) and H G 7in(D). Clearly (6.1) is a bijection. Denote this group by GHn(D). Then parallel to Theorems 3.1 — 3.3 and Definition 3.1 of Chapter 3 we have T h e o r e m 6.1: GHn(D)

acts transitively on Hn(D).

□

Definition 6.1: Let Hi and H2 be two points of Hn(D). The arith­ metic distance between H\ and H2, denoted by a d ^ i , ^ ) ? is defined to be rank(i/i — #2)- If ad(#i,if2) = 1> then they are said to be adjacent. □ Theorem 6.2: Let Hu H2, H3 G 7U(D).

Then

1° a d ( # ! , H2) > 0; ad(#1, H2) = 0 if and only if Hx = H2. 2°

ad(^ 1 ,ft)=ad(J3 r 2 , J ffi).

3° *A{HuH2)+*A(H2,Hz)>*dL{HuH*).

u

T h e o r e m 6.3: The elements of the group GHn(D) leave the arithmetic distance between any two points of 7in(D) invariant. □ However, the set of pairs of n x n hermitian matrices of arithmetic distance r for 1 < r < n does not always form an orbit under GHn(D). This is because

6.1. The Space of Hermitian

Matrices

307

two hermitian matrices of the same rank are not necessarily cogredient. However, if we assume that the map N is surjective, i.e., any a G F can be expressed as a = da for some a G D, then any n x n hermitian matrix of rank r is cogredient to

(/W ^

"l n r 0( - ) J

and, hence, any two nxn hermitian matrices of the same rank are cogredient. It follows that for any r with 1 < r < n, the set of pairs of n x n hermitian matrices for which the distance between each pair is r form an orbit under the group GHn(D). From Theorem 6.3 we know that the arithmetic distance is a geometric invariant under GHn(D), so is, in particular, the adjacency. Our purpose is to characterize the transformations of the form (6.1) by as few invariants as possible. We will see that the adjacency alone is almost sufficient to characterize the transformations of the form (6.1) to within automorphisms of D and scalar multiplications by elements of F*. More precisely, we have the fundamental theorem of the geometry of hermitian matrices, which reads as follows. Theorem 6.4: Let D be a division ring which possesses an involution — and assume that the Assumptions 1° and 2° are satisfied. Let n be an integer > 2. When n = 2 we assume further that D is a field. Then a bijective map A from 7in(D) to itself such that both A and A-1 preserve the adjacency is of the form A(Z) = at~PZaP + H0

for all Z G Hn(D),

(6.2)

where a G F*, P G GLn(D), H0 G 7in(D), and a is an automorphism of D which commute with —, i.e., ~aa —"aF for all a G D. Conversely, any map of the form (6.2) from 7in{D) to itself is bijective and both the map and its inverse preserve the adjacency of pairs of hermitian matrices. □ After some preparations in Sections 6.1 — 6.3 and 6.5, Theorem 6.4 for n > 3 and n = 2 will be proved in Sections 6.4 and 6.6, respectively. Parallel to Definition 3.2, Proposition 3.5, and Corollary 3.6 we have

Chapter 6. Geometry of Hermitian

308

Matrices

Definition 6.2: Let # , H' G Hn(D). When H ^ H', the distance d(H, H'). between H and H' is denned to be the least positive integer r for which there is a sequence of r + 1 points H0, Hi, H2, • • •, Hr with H0 = H and Hr = H' such that H{ and Hi+i are adjacent for i = 0,1,2, • • •, r — 1. When # = # ' , define d(H,H) = 0. Proposition 6.5: For any two points if, H' £ Hn{D)
Corollary 6.6: Let A be a bijective map from Hn{D) to itself and assume that both A and .4" 1 preserve the adjacency, then A also preserves the arithmetic distance in Hn{D), i.e., for any Hi,H2 G Hn(D) rank(i7i — #2) = v<mk(A(Hi) - A(H2)).

6.2

Q

Maximal Sets of Rank 1

Definition 6.3: A nonempty subset M. of 7in(D) is called a maximal set of rank 1 if any pair of points of M. are adjacent and there is no other point of li,n(D), outside A4, which is adjacent to each point of Ai. □ Proposition 6.7: Under a transformation of the form (6.1), a maximal set of rank 1 is transformed into a maximal set of rank 1. D Define Mi = {£Eii\t e F } ,

i = 1,2, • • • ,n.

(6.3)

Proposition 6.8: All Mi's,i = 1,2, • • • , n , are maximal sets of rank 1. Moreover, every maximal set of rank 1 can be transformed under a trans­ formation of the form (6.1) to the set Mi. Proof: Let M be a maximal set of rank 1. Assume that H0 G M.

Then

the transformation Z^+Z-Ho,

(6.4)

which is of the form (6.1), carries M to a maximal set of rank 1 containing the zero matrix 0. Denote the latter by M'. Let Hx G M' and Hi ^ 0.

6.2. Maximal Sets of Rank 1

309

Then Hi is of rank 1. There exists a P G GLn(D)

such that

tpH^P = 6 £ n , where £i G F*. After performing the transformation Z ►—> ' P Z P , which is of the form (6.1) and leaves 0 fixed, M! is carried into a maximal set of rank 1 containing 0 and t>\E\\. Denote the latter by M". Let H G M", H ^ 0, and H ^ £i£ii, then H is adjacent to both 0 and £i£ii, i.e., rank if = r a n k ( # - £ i £ n ) = 1, which forces H G Mi. Therefore M" = .A^. D

Corollary 6.9: Let Hi and # 2 be two adjacent points of Hn(D). Then there is a unique maximal set of rank 1 containing both Hi and i? 2 , which is the set of matrices {{l-QHt+tH,

\UF}.

D

Corollary 6.10: If there are two distinct maximal sets of rank 1 whose intersection is nonempty, then their intersection contains a single point. □ Corollary 6.11: Any maximal set of rank 1 can be put into the following general form { t ( a i , a 2 , - " , a n ) ^ ( a i , a 2 , - - - , a n ) - f H0\£ G F}, where (ai, a 2 , • • •, a n ) is an n-dimensional nonzero row vector over D and Ho is a point of 7in(D). □ Proposition 6.12: Let A4 and M1 be two distinct maximal sets of rank 1 with a nonempty intersection. Then they can be transformed simultaneously under a transformation of the form (6.1) to A^i and .M 2 , respectively. Proof: By Proposition 6.8 we can assume that M. = A4i. By Corollary 6.10, Mi fl M' consists of a single point, let it be called H$. Then the transformation (6.4) leaves Mi invariant and carries HQ into 0. Thus we can assume that Mi H M' = {0}. Let H G M' and H ^ 0. We write rj _ ( tin

u

\

Chapter 6. Geometry of Hermitian

310

Matrices

where hn G F , U is an (n — l)-dimensional row vector over D, and H22 is an (n — 1) x (n — 1) hermitian matrix over D. Since # is adjacent with 0, rank if = 1. But H £ M, so H22 ^ 0 and is of rank 1. There is a Pi G GLn-i(D) such that P1H22P1

= £2^11 j

where £2 G -F*. Then the transformation 1

1 Pi J

V

p

i

leaves Af 1 fixed and carries H to a matrix of the form / a n ai2 Ol2 6 % 0 #1 =

ai3 0 0

\ a£

0

0

0 0 •••

0

/

where a n G .F and ai2,ai3, • • • , a i n G D. Thus we can assume that M! contains Hi. Since rankifi = 1, we must have ai 3 = • • • = a i n = 0 and fln^2 — 012012 = 0. Then under the transformation t

1 -012&"1

1 1

I%

-0126-

1

j(n-2)

1 j(n-2)

which leaves Mi fixed and carries Hi to £2-^22? we can assume that ,M! contains 0 and £2^22- By Corollary 6.9 M' = M2D Clearly, the set

Ml2 = Q(n-2)

UF\

is also a maximal set of rank 1 and Mi C\ M2 H Mi2 have

(6.5) = {0}. Moreover, we

Proposition 6.13: Let M, M', and M" be three distinct maximal sets of rank 1 with a nonempty intersection. Then they can be transformed

6.3. Maximal Sets of Rank 2

311

simultaneously under a transformation of the form (6.1) to Mi, M2, and M3, respectively, or to .Mi, «M2, and M12 respectively. When n = 2, only the latter case can occur. Proof: By Proposition 6.8 we can assume that M = Mi and M' = M2. By Corollary 6.10, Mi H M2 n M" = {0}. Let if G .M" and IT ^ 0. We write 5 =

#11

#H12 12 \

H\2

#H22 22 / '

where Hu is 2 x 2, H12 is 2 x (n - 2), and H22 is (n - 2) x (n - 2). Since ad(if, 0)=1, if is of rank 1. Hence rank #22 < 1- We distinguish the follow­ ing two cases. (a) rank H22 = 1- Similar to the proof of Proposition 6.12, under a transfor­ mation of the form (6.1), which leaves M\ and M2 fixed, H can be carried into £3^33, where £3 G F*. Then M" is carried into M3. (b)

rank #22 = 0. Since rank H = 1, H is necessarily of the form

(

/ Ah\\ n

* =

Al2

hAl2 12

\\

A n ^12^12

Q(n-2) j

A12 h^h12h12 0

Then the transformation tt 11 An Kl X\ ►

I

(-2)

An 7^ 0, Ai2 ^ 0.

,

An^O, Ai2^0.

/

w \

j{n- -2)

,

/' 1

\ hu K221

^

leaves Mi and .M2 fixed and carries H into hu(En Corollary 6.9, .M" is carried into .M12.

6.3

j(nj ( n -.2 ) )

+ E i 2 + £21 + £22)• By n

Maximal Sets of Rank 2

Definition 6.4: A subset C of /Hn(D) is called a maximal set of rank 2, if the following conditions are fulfilled: M. 1° C contains a maximal set of rank 1, denoted by M.. 2° For any H G € C\M

and T €G .M, M, a d ( # , T) = 2.

Chapter 6. Geometry of Hermitian

312 3° For any H G Hn(D),

Matrices

a d ( # , T ) = 2 for all T G M, implies H G £ . □

Proposition 6.14: Under a transformation of the form (6.1), a maximal set of rank 2 is transformed into a maximal set of rank 2. O

Let

( ° ' •• Ci= <

0 • •• Wi • 0 •

I V I 00 •••••■

0

X\i

0

• •• • •

0

%i—l,i

0

%i—l,i

%ii

•£i,z'+l

0

^t,t+l

0

••

00

x— ^t,n

0

.. ••••

0 \ 0

xn G F and #1,-,

•^in

I

5 «^Z —1,2*5 *^i,2 + l 5

0

5 "^m

^

► 5

-^

0 /) 1 < i < n.

(6.6)

Proposition 6.15: All £t-'s, i = 1, 2, • • •, n, are maximal set of rank 2. Moreover any maximal set of rank 2 can be transformed under a transfor­ mation of the form (6.1) to the set C\. Proof: Let £ be a maximal set of rank 2 and A4 be a maximal set of rank 1 contained in C By Proposition 6.8 we can assume that M, = M.\. Let H G C\M\ and write # = {h>ij)l
/in

u

{ % ##22 22 ft

/

where h^ — hji for all i,j = 1,2, • • •, n, w = (A12, A13, • • •, ftin), and H22 — (hij)2
T2

Q(n-3) Q(n-3) 1 >

where T2 is a 2 x 2 invertible hermitian matrix. After performing the trans­ formation t 1' 1 \ ( 1 \ 7 —-> Z .. , \V * / V flj' which leaves M.\ fixed, we can assume that C contains

(I " I

/in f fcn 1*1 = ^

/T2

v

\

1 '

Q(»-3) J /

6.3. Maximal Sets of Rank 2

313

where v = uPi. Since ad (#1,0) = 2, Hi is of the form

#1 =

/( An hi h\2 h\2

hhw 12

hi his 3

\\

rp

T2

^13

0Q( n - 3 )

(n-3) fj

V\

But it is impossible that ad(i?i, XEu) — 2 for all \ € F. Thus the case rank H22 = 2 cannot happen. If rank #22 = 1, then there is a P2 G GL n _i (D) such that P2H22P2 P2H22P2 == ^£^ai> 11,

where £ £ F. After performing the transformation Z>^ ' ( ' * ) • which leaves Mi fixed, we can assume that C contains /An 2fi =

%u2

1

/

t

^

x

I

\ '

V o<"- 2 >) J

where v2 — uP2. Since a d ( # i , 0 ) = 2, # 1 is of the form

( hn

ft-

hi2

hi2

t

\ 0(„-2)

j

Since ( / O w e can find an element A0 G F such that (fen — A0)£ — ^12^12 = 0. Then ad(£Ti, Ao-En) = 1, this is also a contradiction. Therefore we must have rank#22 = 0, i.e., #22 = 0. Hence H G d. □ Corollary 6.16: Any maximal set of rank 1 is contained in a unique max­ imal set of rank 2. The unique maximal set of rank 2 containing Mi is C\.

□ Corollary 6.17: Any maximal set of rank 2 can be put into the following general form ( #11

< tp

^12

\ zi^

#12 * 0

•

•

0 • •

0

0 j

\P + H0

£11 G F , X i 2 , - • • ^ l n ^ ^

►,

^

(6-7)

Chapter 6. Geometry of Hermitian

314

Matrices

where P G GLn(D) and H0 is an element of Hn(D). Any maximal set of rank 2 containing 0 can be put into the following form ( Xl\

#12

tp

X\n

#11 € F,

0 where P G GLn(D).

\

0 X12,'"

, « l n € i?

(6.8)

/

□

The converse of Corollary 6.16 is not true. In fact, a maximal set of rank 2 contains several maximal sets of rank 1. However, we have the following proposition. Proposition 6.18: Let £ be a maximal set of rank 2 and H0 G C. Then there is one and only one maximal set of rank 1 contained in C and containing Ho. Proof: By Proposition 6.15 it is sufficient to consider the case C = C\. Let M be any maximal set of rank 1 contained in C\ and containing Ho- If Ho = 0, then every element of A^\{0} is of rank 1. Therefore M = M.\. If Ho ^ 0, then the transformation (6.4) leaves C\ fixed, carries Ho to 0, and carries M to M — H0. By the preceding case, M — H0 = Mi. Therefore

M = Mi + S0.

□

Proposition 6.19: Any two distinct maximal sets of rank 2 which have a nonempty intersection can be carried simultaneously by a transformation of the form (6.1) to C\ and C2. Proof: Let C and C be two distinct maximal sets of rank 2 which have a nonempty intersection. Let Ho G C D C By Proposition 6.18 there is a unique maximal set of rank 1, denoted by M, which is contained in C and contains if05 and there is a unique maximal set of rank 1, denoted by M', which is contained in C and contains Ho. Since C ^ £', by Corollary 6.16 M 7^ M'. By Proposition 6.12 M and M1 can be carried simultane­ ously by a transformation of the form (6.1) to M\ and M2, respectively. Then by Corollary 6.16 C and C are carried simultaneously to C\ and £ 2 , respectively. □

6.3. Maximal Sets of Rank 2

315

Proposition 6.20: Let 1 < i < j < n. There exist exactly two maximal sets of rank 2 which contain the set {xE^ + xEji\x G D} in common. More precisely, they are d and Cj. Proof: It is enough to consider the case (z, j) = (1,2). The other cases can be treated in the same way. It is clear that C\ and C2 axe two maximal sets of rank 2 which contain the set {xE12 + xE2\\x € D} in common. Now let C be any maximal set of rank 2 containing the set {xE\2 + xE2\\x G D}. By Proposition 6.18 let M be the unique maximal set of rank 1 contained in C and containing 0. Let H G M and H ^ 0, then rank if = 1. Write H in the block form H = ( ^ n ^12 I \ '#12 H22 J ' where Hn is 2 x 2, Hi2 is 2 x (n - 2), and H22 is (n - 2) x (n - 2). From rank if = 1 we deduce rank Hn < 1. We distinguish the following cases: (a) i / n = 0. From rank if = 1 we deduce H\2 = 0 and ranki/22 = 1- Then H — (£"12 + £21) is of rank 3, thus ad(if, E12 + E2\) = 3, which contradicts to the assumption that £ is a maximal set of rank 2. Therefore the case Hn = 0 cannot occur. (b)

Hn 7^ 0. Then rank #11=1. There are only three possibilities: Hll=

{

0j '

(

h) '

( k h-'kk ) '

In the first two cases h ^ 0 and in the third case hk ^ 0. For the first case, from rank if = 1 and ad(H,E12 + E2i) = 2 we deduce H = hEn, then by Corollary 6.9 M = Mi and by Corollary 6.16 C = C\. For the second case, we have similarly £ = £2. These two cases are what we want. It remains to prove that the third case cannot occur. Suppose that H l l =

\ k

h^kk

) '

where hk ^ 0. By Corollary 6.9 we can assume that A = 1. From rank H = 1 and ad(if, Ei2 + E21) = 2 w e deduce

fii

\

k kk

H=\

I

.

0 /

(6.9)

Chapter 6. Geometry of Hermitian

316

Matrices

Let P = I + kEw. Then the transformation 1 z - -» ^p-^zip- )

carries H into .En. By Corollary 6.9 there is a unique maximal set of rank 1 containing 0 and En, which is Mi, and by Corollary 6.16 there is a unique maximal set of rank 2 containing Mi, which is Ci. It follows that M = tPMiP and C — tPCiP. Then C consists of elements of the form fcxn + xn ^13 xnkk + x~[2~k + kxi2 kxi 3 fcxn + x~^i x~^k 0 ^13

( _

-Iff

\

Xn

Xink

Xin

*

0

*

• • ■

Xin KXin

•• ■

^

0 0

,

(6.10)

,

where xn G F and £12, • • •, xin G D. Denote the element (6.10) by M{xn,xi2, - • - , # i n ) . By hypothesis, for any x G D, xEyi + ~xE2i G C, thus there are elements 6 1 G F and 62, • * * ? 6 n G -D such that xEi xEi22 + + xxE£22i1 = A Aff((661i , 6 a2 , ••' •* ,fm). ,6n)« Consequently, 6 1 = £13 = • • • = 6 n = 0, £12 = x, and xk + kx = 0. Let x = 1, we obtain k + k = 0. Thus x~k — kx = 0. Since fc ^ 0, we have of = kxk'1 for all x £ D, which implies that D is a field. Then x = x for all x £ D, which contradicts Assumption 1°. □ Let 1 < i,j < n and i ^ j . For any k G D, define dj(k)

= ' ( / + kEiA&il

+

kE{j).

(6.11)

Clearly, when k runs through D, £tj(fc)'s are distinct maximal sets of rank 2 andA-^O) = &. Proposition 6.21: Let 1 < i < j < n, x \—► xu(x) be a map from D to F for which xn(0) = xn(l) = 0 and xu(x) ^ 0 for some x G D*, and {a;,-,(a;)E,t- + xEij xE^ + xEji\x G !>}. Sij = {xn{x)En D}. 5ii = Let k G D be such that fc + A; = 0 and xu(x)k2 = kx — xk for all x G JD, then Aj(fc) is a maximal set of rank 2 containing <Stj. Conversely, a maximal set

6.3. Maximal Sets of Rank 2

317

of rank 2 containing Sij is of the form Cij(k) where k € D satisfies k + k = 0 and xu(x)k2 — kx — xk for all x G D. Proof: The first statement can be easily verified. Now let us prove the second statement. As in the proof of Proposition 6.20 it is enough to consider the case (i,j) = (1,2). Let C be any maximal set of rank 2 containing <Si2Since 0 G <Si2, by Proposition 6.18 there is a unique maximal of rank 1 contained in C and containing 0; let it be M.. Let H G M. and H ^ 0, then rank H = 1. Write H in the block form Hl2 H = ( Hn | \ H12 H22 ) '

where Hn is a 2 x 2, Hu is a 2 x (n — 2), and H22 is an (n — 2) x (n — 2) matrix. From rank if = 1 we deduce rank Hn < 1. As in the proof of Proposition 6.20, we can show that the case r a n k i / n = 0 cannot occur. Therefore rank Hn = 1. There are three possibilities:

^n = (

oj'

(

&)'

(if

h-'kk)-

In the first two cases h ^ 0 and in the third case hk ^ 0. For the first case, from rankH = 1 and ad(if, £12 + #21) = 2 we deduce H = hEu, then M. = M.\, £ = Ci = £12(0), and C 3 5 i 2 . For the second case we have H = hE22, then M. = A42, C = C2, but C2 2 ^12; therefore this case is impossible. Now consider the third case. By Corollary 6.9, we can assume that h = 1, i.e., Hn =

[k kk)>

where k ^ 0. As in the proof of Proposition 6.20, we can prove that H is of the form (6.9) and let P = I + kE\2 we can prove that C = tPC\P and C consists of elements of the form (6.10). Denote the element (6.10) by M ( x n , #12, • • •, xin).

By hypothesis, for any i G J ) , £ I I ( # ) 2 ? I I + xE\2 +

~x~E2\ G £ , thus there are elements £n G F and £12, • • •, £i n G D such that z i i ( z ) £ n + xE12 + xE21 = M(£n,£i2, • • • , 6 n ) . Consequently, i n ( i ) = fn, x = £uk + 6 2 , 6 1 * * + 6 2 ^ + ^ 1 2 = 6 3 = ••• = £i n = 0. Let x = 1, then we deduce k + k = 0. It follows that Xn(x)k2 = kx — ~xk. □

Chapter 6. Geometry of Hermitian

318

Matrices

It is clear that the maximal set of rank 2 C\ has an additive group structure, that is, C\ is an abelian group with respect to the matrix addition. More generally, we have Proposition 6.22: Let C be any maximal set of rank 2 containing 0. Then C is an abelian group with respect to the matrix addition, and the maximal set of rank 1 contained in C and containing 0 consists of those matrices of C which are of rank < 1 and is a subgroup of C. Denote this subgroup by M. Then any coset of C relative to M is a maximal set of rank 1 and conversely any maximal set of rank 1 contained in £ is a coset of C relative to M. Proof: Since 0 G £ , by Corollary 6.17 C can be expressed in the form (6.8). Then it is obvious that C is an abelian group with respect to the matrix addition and

M =

/ xn 0

0 ••• 0 \ 0 ••• 0

V 0

0

xneF

(6.12)

0/

is the maximal set of rank 1 contained in C and containing 0. Clearly M consists of those matrices of C which are of rank < 1 and M is a subgroup of C. It is easy to verify that any coset of C relative to M is a maximal set of rank 1. Now let M1 be a maximal set of rank 1 contained in C. Let H0 be an element of M', then M' — H0 is also a maximal set of rank 1 contained in C. Clearly, 0 e M' - H0, hence M' - H0 = M and M' = H0 + M is a coset relative to M. □ Definition 6.5: Let £ be a maximal set of rank 2 containing 0 and M be the maximal set of rank 1 contained in C and containing 0. Then the set of elements of the factor group C/M is called the reduced maximal set of rank 2 of £ and denoted also by C/M. Any complete system of representatives of the cosets of C relative to M is called a system of representatives of C/M.

□

6.3. Maximal Sets of Rank 2

319

For example, the set

{1 I //

0

X £12 12 00

*12

•'"Xin ••• •••

\|

^

00

,xlne

Z12,"

•••

0

\ ^T

D\

0/

. Ui2, -,xlneD\ is a system of representatives of the reduced maximal set of rank 2 C\jM\ of £ 1 . More generally, the set is a system of representatives of the reduced maximal set of rank 2 C\jM\ •v Z l l ( Z l 2 , "the • ,Xset X12 ' * Xin \ ln) of L\. More^generally,

... 0/1

\x^ 0

J

■• 0 ' "• X12

0 Xin\\

«12

0

0

1\

: ^T^

0: • ••

[\

x~^

0 — 0/1

<(

I X n ( X i 2^12 , * ' * , Xln)

••

D \1

•>xlne

^12,•'

12 Xln 0: / p ' " ' >

e D

(

J

where #11(212, • • •, X\n) is an element of F depending on £12, • • •, xin, is also a system of representatives of the reduced maximal set of rank 2 C\jM.\ of A . Now let us study the intersection of two maximal sets of rank 2. Let C and £ be two distinct maximal sets of rank 2, and assume that C fl £ 7^ (j>. By Proposition 6.15 we can assume that C = C\. Let HQ G C\ fl £. Then the transformation (6.4) Z ^1— Z> -Z H- Q H0

leaves C\ invariant. Denote the image of £ under (6.4) by £'. Then 0 G C\ fl £". By Proposition 6.18 there is a unique maximal set of rank 1 contained in £' and containing 0, which will be denoted by M!'. By the proof of Proposition 6.12 we can assume that

4 M" =

7( A \ V l

>

1

/("-2) ,f)

where Pi G GLn-X{D) <-

r=

/7i

A 1

^

\-A

1

/M

f P*< j' 1

\

A 1

■M2

J

P

i^

\

-1

j(»- 2 ) y

'1

( \

>

-1

11 ,

and A G £>. By Corollary 6.16,

r\ < / < - » > ,/

l

,

>r ^

A,

/' 1 A A 1

i

_ 1

)(v \

/(-») ,

/ i

>v - i

/

.

Chapter 6. Geometry of Hermitian

320

Matrices

Denote the first row of Px * by (p 2 , • • • ,Pn). Then (p 2 , * * * ,Pn) 7^ (0, • • •, 0) and f / —^12^ — AX12

d n c" = I

IV

X12P2

Xl2Pn

P2^12

0

0

Pn #12

0

0

\

xu € D /

Therefore we have Proposition 6.23: Let C be a maximal set of rank 2 and assume that d ^ C and d D C ^ . Then d fl £ ' is of the form / —#12^ — Axi2

£in£' =

I\

X12P2

X\2Pn

P2~X~T2

0

0

Pn ^12

0

0

\

+ H0 x12 e D /

where A,p 2 r * • >>Pn € -D, (P2, • • * ,Pn) 7^ (0, • • •,0), and H0 is an element of

d n £'.

□

Definition 6.6: Let £ be a maximal set of rank 2 containing 0 and M be the maximal set of rank 1 contained in C and containing 0. The image of the intersection of C with any maximal set of rank 2 distinct from C and having a nonempty intersection with C under the natural map C H

C/M H + M

is called a line of the reduced maximal set of rank 2 C/M

of C.

□

Now we assume that n > 3. From Proposition 6.23 and Definition 6.6 we deduce Proposition 6.24: Let C be a maximal set of rank 2 containing 0, which is of the form (6.8) and M be the maximal set of rank 1 contained in C and containing 0, which is of the form (6.12). Then the map from the reduced maximal set of rank 2 C/M to the (n — l)-dimensional left affine space

6.3. Maximal Sets of Rank 2 AG'(n-l,D)

321

over D

(

0 x^

\ x ^

x12 0

C/M • • • xln ••• 0

AG'(n-l,D) \

P+ M

(X12, • • • , Z l n )

0 /

0

is a bijective map which carries lines of C/M and the map

into lines of AGl(n — 1,JD),

AGr(n-l,D)

C/M (

0

Zi2

x12

0

\ xln

0

• • • Xln

•••

0

\

P+ M zi7T /

0 )

is a bijective map which carries lines of C/M into lines of the (n — 1)dimensional right affine space AGr(n — 1,1?) over I?. □ Denote by Sn(F) the space of n x n symmetric matrices over F. Clearly, Sn(F) C Hn(D). If we regard Sn(F) as a subset of Hn(D), then the arith­ metic distance is equal to the distance, even when F is of characteristic two. Hence we may define maximal sets of rank 1 and of rank 2 in Sn(F) by Def­ initions 5.3 and 5.4 of Chapter 5, respectively, and prove Propositions and Corollaries of Sections 5.2 and 5.3 in the same way. Let C\ be the maximal set of rank 2 in Hn{D) defined by (6.6) and let

£1(F) =

£1nSn(F).

Then

A(F) =

f / #11 X12

#12 0

Xln \

0 £ll> #12, * ' * ,Xln £ F

[\

Xln

0

oJ

is the maximal set of rank 2 in Sn(F) defined by (5.7).

Chapter 6. Geometry of Hermitian

322

Matrices

Proposition 6.25: Let C be a maximal set of rank 2 in Hn(D) and assume that d ± C and d(F) fl C > 2. Then &(F) H C is of the form f / 2/1221 2/12^2 • • • yi2qn \ y12q2 0 ••• 0

&(F)nc' = {

I \ yinqn

0

•••

0

+ H0 yi2eF

/

where qu q2, • • • ,qn G F, (q2, • • •, tfn) ^ (0, •, 0), and Ho is an element of

&(F)nc. Proof: In the proof of Proposition 6.23, which was given above that propo­ sition, if we replace C\ by C\(F) then we can prove that C\(F) C\C consists of matrices of Sn{F) of the form / "^12^ — A x\2 X12P2 P2^12 0

V

0

Pn ^12

Xl2Pn \ 0 + H0, 0

(6.13)

)

where Ho is a fixed element of £ i ( F ) , A,p2? • * • ,Pn are fixed elements of D, {P2, • * • ?Pn) 7^ (0, • • •, 0) and xi2 runs through those elements of D, such that all X12P2? • • * 5#i2Pn are elements of F. If xi2 = 0, then (6.13) becomes Ho. Since \d(F) D C'\ > 2, there is a £12 G Z>* such that &2P2, • • • ,£ 1 2 p n G F. Let gi = - 6 2 A - A ^12, ? 2 = 62P2, • • •, qn = £\2Pn, then ?i, q2, • • •, gn G F and (#2, • * •, qn) 7^ (0, • • •, 0). For any xX2 G D* such that Zi 2 p, • • •, x12pn G F , let 2/12 = ^ 1 2 ^ 5 then yi 2 G F* and (6.13) can be expressed as / 2/1231 ^12^2 • • •, yuqn \ y12q2 0 ••• 0 + #0. \ yinqn

0

(6.14)

0 J

Conversely, it is clear that any point of the form (6.14), where yi2 G F can be written as a point of the form (6.13). □ Definition 6.7: Let £ x be the maximal set of rank 2 of Hn(D) defined by (6.6). Then a line in C\(F) is either a maximal set of rank 1 contained in C(F) or the intersection of Ci(F) with any maximal set of rank 2 of Hn(D) when it contains at least two points.

6.4. Proof of the Fundamental Theorem (the Case n>3)

323

From Proposition 6.25 and Definition 6.7 we deduce Proposition 6.26: The map

A(F)

—+

AG{n,F)

/ xn X\2

0

•••

0 1

X\2

V Xin 0 \

•••

>

( z i l , X i 2 , • • ■ ,3>ln)

0 J

x

ln

is a bijective map which carries lines of C\(F) into lines of the n-dimensional affine space AG(n,F). More generally, similar to Definition 6.7 and Proposition 6.26 we have Definition 6.8: Let d (1 < i < n) be the maximal set of rank 2 of /Hn(D) defined by (6.6), and &(F) = & H Sn{F). Then a line in &(F) is either a maximal set of rank 1 contained in Ci(F) or the intersection of Ci(F) with any maximal set of rank 2 of Hn(D) when it contains at least two points. □ Proposition 6.27: The map

—»

d(F)

( °'

••• ••

0

xu

0

• •

0 ••••

0

Xi-l,i

0

• • 0

I 0

I

X\{

t-l,t ' •* •* Z «^t—l,t

0 ••••

0

■• 0 •••• V\ o

0

X%i Xi,i+\

X%n

X%n

0

•

0

• •

■

AG{n,F) AG(n,F)

0 \

0

1

^

^ l t ? * * ' 5 X{—i)t", Xu, *^t,t+l?

* * ? Xin)

0 )

is a bijective map which carries lines of Ci(F) into lines of AG(n,F).

6.4

□

Proof of the Fundamental Theorem (the Case n > 3)

Now we assume also that n > 3. The second statement of Theorem 1 is obvious. We give the proof of the first statement in the following.

Chapter 6. Geometry of Hermitian

324

Matrices

Let , 4 b e a bijective map from Hn(D) to itself such that both A and A * preserve the adjacency. By Corollary 6.6, A preserves the arithmetic dis­ tance, i.e., for any HUH2 G W»(fl), ad{H1,H2)=ad{A{H1),A{H2)). Then by Definitions 6.3 and 6.4 A carries maximal sets of rank 1 to maximal sets of rank 1 and maximal sets of rank 2 to maximal sets of rank 2. We proceed by steps to show that after subjecting A to bijective maps of Hn(D) of the form (6.2) successively, A has the form A(Z) = Z

for all Z G Hn(D),

then the first statement of Theorem 1 follows. (i) At first, after subjecting A to the bijective map Z h-+ A{Z) - .4(0)

for all Z G Hn(D)

of /Hn(D)J we can assume that A(0) = 0.

(6.15)

M.\ and Ad2 are two maximal sets of rank 1 whose intersection is {0}, so are A(M\) and A(M2). By Proposition 6.12 after subjecting A to a bijective map of /Hn{D) of the form (6.1), we can assume that A(Mi)

= Mi,

i = 1,2.

(6.16)

C\ and C2 are maximal sets of rank 2 containing M.\ and M.2, respectively, so are A(Ci) and A(C2). By Corollary 6.16 and (6.16) we must have A{h)

= Ci,

i = 1,2.

(6.17)

Then A induces a bijective map A on the set of maximal sets of rank 1 contained in £ 1 ? i.e., on the reduced maximal set of rank 2 C\jM\. By Definition 6.6 A carries lines of £i/Mi into lines of Ci/Mi. By (6.17) we can assume that /

0 xU

x12 0

X\n

0

\

/

X 11 c

X

X

12

''*

0

•••

0

0

•••

0

ln

\

12

V*ln

/

6.4. Proof of the Fundamental Theorem (the Case n > 3)

325

for all £12, • • •, #i n € D, where x\{ = x^x^, • • • , £i n ) is a function of £12, • • *»#in with values in D for all i = l , 2, • • •, n and ^ ( x ^ , •, x\n) G P . The set ( (

0

«u

I \x^

X12

' ' '

Xi n

\

o ... o

0 •••

0/1

is a system of representatives of the reduced maximal set of rank 2 £ i / A 4 i , so is the set #ii_

#12

x\2

0

0

V*ln

0

o J

I /

*ln\

x

i\ = Xn{x12,"-,xln) G F, *it = *it(zi2, *' *, xin) € £> for i = 2, • • •, n, where xi 2 , • • •, £i n G -D

By Proposition 6.24, the fundamental theorem of affine geometry (i.e., The­ orem 2.7), and (6.15), we can assume that (# 1 2 ,

* * * , Xln)

(#12,

=

• • • , X\n)

Py

where cr is an automorphism of D and P G GLn-i(D). From (6.17) we deduce A(& H £ 2 ) = £ i H £ 2 . Clearly, £ i n £ 2 = {xEu + xE2i\x G D}. It follows that P is of the form P22

P =

<** Pn-2 I '

where p22 G D*, w is an (n - 2)-dimensional row vector, and P n _ 2 G GLn-2(D)-

After subjecting .4 to the bijective map of Tin(D)

-M^M"' which leaves 0, Mi, M2, £ i , £2 invariant, we can assume further that /

0

x12

xTJ

0

0

0

0

\~xul

Xln

\

/

I

#11

#12

xh

0

0

0

0 y

\*L

(6.18)

326

Chapter 6. Geometry of Hermitian

for all x12, • • • j i r i n G A where x*r = x^(x12,

• • •, xln).

Matrices

Since »4(A fl C2) = <

£ i f l £ 2 = {^#12 + xE21\x G JD}, we have A{xE12 + xE21) = x°El2 + x~*E21 for all x G D.

(6.19)

From (6.18) we have A(xE13 + xE31) = x*u(0, x, 0, • • •, 0)En + x"E13 + x^E31. By Assumption (ii) there is an element a G D such that ^ ( 0 , 1 , 0 , • • •, 0) = a + a. Then after subjecting A to the bijective map

Z H-» \I-aE3l)Z{I

- aEsi),

which leaves 0, A4i, M2, £ i , C2 invariant, we have (6.18) with # ^ ( 0 , 1 , 0 , • • •, 0) = 0. If arJ^O, x, 0, • • •, 0) = 0 for all x G D then A(xE13 + xE31) = xaE13 + ~x*E3X for all x e D.

(6.20)

Now consider the case #n(0, x, 0, • • •, 0) ^ 0 for some x G D . By Proposition 6.20 there are exactly two maximal sets of rank 2 containing {xEi3-\-xE3i \x G D}. Thus there are exactly two maximal sets of rank 2 containing {A(xE13

+ xE31)\x eD}

= {xJ^O, x, 0, • - •, 0)EU + xaE13 + x^E31 \x G D } .

Clearly, one of them is C\. By Proposition 6.21 the other one is necessarily of the form £13(^3) where A;3 G D* satisfies k3 + k3 = 0 and #n(0, #, 0,•••, 0) = xak^ - k^x* for all x G D. Then ^(a:E 1 3 + xE31) = (a;*^ 1 - k^x^jEn

+ xaE13 + x*E3i

for all x e D.

After subjecting A to the bijective map

Z .—► '(I-k^E^Zil

- *£^si),

which leaves 0, A^i, .M2, £1? £2 invariant, we can assume that (6.15) — (6.20) hold. In a similar way, we can assume that (6.15) — (6.20) hold and A{xExj + xEfl) = xaExj

+ x^Ej!

for all x G D, j = 4, • • •, n.

6.4. Proof of the Fundamental Theorem (the Case n>3)

327

Hence AfaEu

+ xEfl) = x*Eu + x^En

for all x E D, j = 2,3, • • •, n.

(6.21)

It follows from Proposition 6.20, (6.17), and (6.21) that A(Cj)=Cj,

j = l,2,-..,n.

(6.22)

Therefore A{C{nCj)

= CinCj,

i
(6.23)

In particular A(Eij + Eji) = x\fEi:} + x[fEji

for some x\f € D, 1 < i < j < n. (6.24)

(ii) We have ^4(0) = 0, A(C2) = £2? and A(M2) = M2. We can assume similarly /

\

0

x12

0

#12

0

X23

0

5

0

0

x2n

0

0 * * *

/

\

^2n

0

x*2

^12

••• 0

0

V0

0/

x

22

^

3

0 x

•••

23

0

\

^2n

0

X

0

*n 0

for all X12, z 2 3, • • •, a?2n € -D, with x£2 = x*12(x12, z 2 3 , • • •, x 2 n ), »5.- = #23, * • * 5 ^2n) for i = 2,3, • • •,ra,and ( x 1 2 5 X 23> ' ' ' 5 X2n)

— ( X 1 2 5 ^23? * * * > # 2 n )

^{xu,

(6.25)

^25

where a2 is an automorphism of D and P 2 G GLn-i(D). Substituting (1, 0, •••, 0), (0, 1, 0, ••-, 0), ••• (0, 0, •••, 1) successively into (6.25), we deduce from (6.19) and (6.24) that P2 is a diagonal matrix of the form [l,p 3 ,p 4 , * * * ,Pn], where p3p4 • • -pn ^ 0. Therefore /

0

x12

0

0

^12

0

z23

^2n

0

%

0

0

\

\ o % o ••• o ;

/

0 V\2

xtf* ^22

0

0

23P3

*2nPn

0

o

\ o xa22npn o

0

0

xgps

\

•v.^2 » ,

X

/

Chapter 6. Geometry of Hermitian

328

Matrices

For n = 3, we write p 3, the arithmetic distance between / 0 0 1 ••■ 1 \ 0 0 0 ••• 0 1 0 0 ••• 0

/ 0 0 0 ••■ 0 \ 0 0 1 ••• 1 0 1 0 ■•• 0

and

0

\ 1 0 0

0/

Vo i o

is equal to 2; from which it follows that p3 = p4 =

pn.

Let p2 = P3,

then P2 € D* and (

\

0

x12

0

0

^12

0

£23

#2n

0

xS

0

0

0

x^

0

/

\

o ^li*

0

#12 2

^22

Z23P2 23^

0

^23P2

0

V o

*2>2

0 \ 2,lP2 In. 0

x

0

/ (6.26)

Similarly, we have /

0

0

sis

0

0

0

0

X23

0

0

0

0.

034

0

0

V 0

0

x^

0

0

\

X3n

0 0 ^IT 3 0 0

0 0 X23T3P3

xli* 3 ^23a3P3

0 0

/ 0 0

33

Z34P3

0

^34^3

0

0

0

*3>3

0

0

x

• •'

*3>3

(6.27)

where p 3 G D*, etc. We assert that a2 = a3 - • • • = crn, pt- = p ^ f (t = 2, • • •, n — 1), and ^2^2 for all x € D.

(6.28)

6.4. Proof of the Fundamental Theorem (the Case n>3)

329

Consider the images of / O x x 0

/ 0

\

X

x Q(n-3) J

V

\

/0

0

and

0 Q (n-3)

\

J

\ 0 x x 0

o(n-3) y

\

where x G £ , under A By (6.18), (6.26), and (6.27) we have xc = x*2, x° = x a 3 , and xa2p2 = xa3p3

for all x G D.

It follows from the first two equations that a2 = 03. Similarly, 0*3 = <74 = • • • = crn. The third equation now becomes x°2p2 = ~xa2pz for all x G D. Substituting x = 1 into this equation, we obtain p2 = £3. Similarly, we have pi = p^jTT (2 = 3,4, • • •, n — 1). Then (6.28) follows. Our assertions are completely proved. (iii) For x u G F*, from (6.17), (6.18), and the adjacency of /

0

x12

*12

0

0

\ xin

0

0

/ £ll

and

{x^

)

#12

#ln

0

0

0

0

*12

\

)

we deduce / Z11

£12

Z12

0

0

^ sin"

0

0

/

/ xn x\2

x 12 0

\*?»

0

*?»\ 0

/

for all x u G F, x i 2 , • • •, Xin G A where x\x = x ^ x u , Xi2, • • •, xln) G F . In particular, / Z11

/ IJJ

X12

X12

0

0

V am

0

0

for all i n , £12, • • • > %\n € .F.

/

Xl2

x\2

0

0

Wn

0

0

(6.29) )

Chapter 6. Geometry of Hermitian

330

Matrices

We distinguish the following two cases. (a) F ^ F 2 . By Proposition 6.26, the fundamental theorem of affine geometry, and (6.15), we have (

*


(*11, *12> * "' i *^ l»nxln) ) =

•,^rPi,

= (X1U *12, • • • , Xln)aiPU (^11^12,-"

(6.30) (6.30)

where <j\ is an automorphism of F and Pi G GLn(F). Substituting (1, 0, •••, 0), (0, 1, 0, •••, 0), •••, (0, 0, •••, 0, 1) successively into (6.30), we deduce from (6.16) and (6.21) that Pi is a diagonal matrix of the form Pi = [c, 1,1, • • •, 1], where c G P*. Substituting (0, x, 0, • • •, 0), where x G P , into (6.30), we obtain xai = x° for all x G F. After subjecting A to the bijective map of the form (6.2) ZH-

- ■ (

H

1 Z n c ' " - 1 .) I/ cl c/c/-0'

,

we have (6.15) — (6.18), (possibly with another a ^ ) , (21) — (24) (possibly with another x\j ), (6.26) (possibly with another p 2 ), (6.27) (possibly with another P3), etc., and A(X) ===X° A(X) Xa for all X G € d(F). d(F).

(6.31)

We assert that <J2 = 03 = • • • = crn = cr, ~x° = x~° for all x G D, and Pi — Pz = •' • = Pn = 1. For simplicity of writing, we write down only 3 x 3 submatrix situated at the upper left corner of an n X n matrix if all other entries of the matrix are zero. We have proved in (ii) that cr2 = cr3 = • • • = ^Vij Pi — Pi+i {i = 2,3, • • •, n — 1) and deduced the equation (6.28). It follows from Assumption 1° that there is an a G D\F. Let A = a — a, then A G D* and A + A = 0 . The arithmetic distance between

0 1 1\ 1

U

/

and

/° 0 A\ I

X <W

is 2. By (6.31), the first matrix is left by A. By (6.23) and (6.26), ,4

/°

0 A

\ = ,f °

V x oj

\

0

A>2

\.

X"2P2

0

/

6.4. Proof of the Fundamental Theorem (the Case n > 3)

331

Thus / Ou 11 rank j 1 _00 rank \I 1 -\°*V -\°2p22 v

1 A

\

a2

--X A C Tp2 2P 2

1 ==22.-

0

/

It follows that \
M

-£ o\

"*

J

V 0

/°0

and

\'

«

V

0;

we deduce

f° ^

*

V

^ = / o r >1 forall£GF.

oj

*)f

V

Choose x G -D such that # + x = 1. From the adjacency of

(°

0

x

V

-

1/

\

/°

and

\ -XX

V

0)

(6.27), p2 = P3 G .F, cr3 = cr, and X*7 = x a , we deduce

I' 0

A

\,

\ 0 x x 1

=

(°

\

I

0 x*p 2

and

/°

xaVi

■

A J

The arithmetic distance between

/o

1 1\

1

Vi

J

0

V

x

^ \

* 1/

is 2, so is the arithmetic distance between

/0

1 1\ and

1

Vi

/

(°0 V

\ 0

x'pt

^ v\ )

■

Chapter 6. Geometry of Hermitian

332

Matrices

It follows that /?2 = 1After subjecting A to the bijective map Z \—► Z*'1, which is of the form (6.2), we also have (6.15) - (6.17) and (6.22), and (6.18) becomes /

0

x12

V *i^

Xin

0

\

0

\ x^

0

0

0 /

V x^

0

o )

(6.32)

for all X12, • • •, xin € D, (possibly with another xj^), (6.26) becomes

( ^u ° 0

^o

(°

xX2

0

Xl2

^22

^23

0

• • 0 ^ • • x2n • ■ 0 =

0

^23

0

• • 0 • • x2n • ■ 0

0

• ••

{o

^

0

• •

Zl2

0

0

^23

^23

Xfri

0

j

(6.33)

0

(possibly with another x^), (6.27) becomes /

\

0 0 0 0 XI3" x l i 0 0 0

0

sis 0 x23 0 0 x 34 xi 0

••• ••• ••• •••

0 \ 0 x3n 0

/

S

...

0 /

\

0

0 0

0 0

sis x23

0 0

0 0

£~13

#23

£33

^34

X3n

0

0

X3J

0

0

0

0 x 5 ^ 0

\

0 J (6.34)

(possibly with another X33), etc., and (6.31) becomes

A(X) = X

foia&XeCi(F).

Furthermore, we assert that A(X) = X

for all X € £ t ( F ) , i = 1,2,3, ••• ,n.

(6.35)

The case i = 1 has just been proved. Now we consider only the case i = 2, since the other cases can be treated in a similar way. Clearly, for all

333

6.4. Proof of the Fundamental Theorem (the Case n > 3) £i2, Z22, • • •, x2n G F , we have

(°

&12

0

X12

^22

#23

0

• • 0 \ • • x2n • ■ 0 —

0

^23

0

• • 0 • x2n • • 0

0

• •

Io

^2n

0

• •

(°

x12

0

Zl2

x 22

£23

0

^23

Vo

^2n

o t

*

0 /

By Proposition 6.27, the fundamental theorem of affine geometry, and (6.15), we have ( X i 2 , Z 2 2 , X 2 3 , • • • , X2n)

X2n)T2Q2,

= (Sl2, ^22, #23, * **,

where r 2 is an automorphism of F and Q2 G GLn(F). We deduce imme­ diately that Q2 is a diagonal matrix of the form Q2 = [1,#2,1 ? * * * 51] and r 2 = 1. From the adjacency of —x x x

I

and

0 Q( n " 2 )

Q(*- 2 )

we deduce g2 = 1. Thus A(X) = X for all X G C2(F). (b) F = F 2 . Recall that we have (6.15) - (6.24), (6.26), (6.27) etc., with cr2 = cr3 = • • • = crn and pi = pT+T (i = 2,3, • • •, n — 1). From (6.15) and (6.22) we deduce A(Mi) = Mi (i = 1,2, • • •, n) and 4 ( £ t t ) = En, i = l , 2 , . . . , n .

(6.36)

By (6.21) we have 4(F l t - + En) = F lt - + £ t l , * = 2,3, • • • ,n.

(6.37)

Then by (6.29) and (6.37) we have A(Eu + Exi + E{1) = En + Eu + Eiu

i = 2,3, • - •, n.

(6.38)

We assert that A(Eu + Ejj) = Eu + Ejh

l

(6.39)

Chapter 6. Geometry of Hermitian

334

Matrices

Let

A(En + Ejj) =:

5Z

X

klEkh

l
where x\k = x~ki (1 < &, / < n). Since En + Ejj is adjacent with En and Ejj, from (6.36) we deduce A(En + Ejj) = xn(En + Ejj) + XijEij +x~]Eji. Assume that xn = 0, then A(En + Ejj) = XijEij + xTjEjj. En + Ejj is adjacent with Eij + Eji and by (6.24) AjEjj + Ejj) = x\f Ejj + affl Ej{ for some x\f G D, so we must have rank((xij - x\f)Eij

+ (xij - x\y)Eji)

= 1, which is impossible.

Therefore xn = 1 and A(En + Ejj) = £,-,• + 25^ + rc^^j + x~]Eji. From the adjacency of En +

JBJJ

and En we deduce x,j = 0. Hence we have (6.39).

Next we prove that

A(EU + Eu + Ea + Eii) = Elt + Etj + Ea + Efl A(Eun + + Eu + En A(E Ea + Eji) En) = = En EU + EU + E\j Eu + En E« + + EjEn Elj + En &i + Eij

} (6.40)

for all 1 < i < j < n. We consider the case i = 2, j = 3 as an example. For simplicity of writing we write down only the 3 x 3 submatrix situated at the upper left corner of the n xn matrix under consideration when all the other entries of the matrix are 0's. By (6.29) we have

xu /f xn //fo0 i1 1i \\, 1 = \ 1 \. 1 J' V i

*(l Hi A

11 11\\

)' 5

/

where xu = 0 or 1. Assume that xu = 1. The arithmetic distance between £ i 2 + £ i 3 + £ 2 i + £ 3 i and E22+E33 is 2 and by (6.39) A{E22+E33) = E22+E33. It follows that f 1 1 1) j ■11 11 1 \ = 2, rank = 2, rank 1 1 1

1 J W

. 11

which is absurd. Therefore

f (0 1 1 \ / 0 1 1\ 1 A = \ 1 \ \. 1 ) Vi )

Consequently

If

A, i \

\ \ i \

=

i

J

/i 1 1 \ *

Vi

/

6.4. Proof of the Fundamental Theorem (the Case n>3)

335

The general case (6.40) can be proved in a similar way. Then we prove that p2 = p3 = • - - = pn = 1. It is enough to prove that p2 = 1. Since E23 + E32 eC2D £ 3 , by (6.26) and (6.27) we have /° (°

A

f

A\ \ ,

V

(° \ (°

\ i 0 1 = 0 1 = i o 'V

1 0/

\

o0 n • p2

Pi 0 / T* 0

V

The arithmetic distance between l l\ (0 0i 1 1 \ / 0 0 and 10 1 and Vi ) 1 / V _ — Skinr*^ JP -- UT« -ixro n a v p 1 is 2, we deduce p2 — 1p2. Since F = F2, we have p2

(°

I

\ 1 01 /

0

— 1

= 1.

Similar to the proof of (6.37) and (6.38) we can prove AiEii +E Eji) =- E^ + En A{E{j + j{) = E{j + Ej{ and A(Eu Eu + + Eji Eji + E^ Eji) ~Eii + A(Eu + Eij + Eji) == En Eij + for all i ^ j , and similar to the proof of (6.40) we can prove + Eik + Eji + Eki A(Eij Eik + + Eji Eji + Eti) --= E^ + A(Eij + + Eik + Eki) = E^ Eik + Eji + Eki and Eu + Eij + Eik + Eji + Eki A(Eu + Ei Eik + + Eij Eij + Eji + Eki) === En A(Eu + + E{j + Eik + Eji + Eki k + Eji + E^ for distinct

i,j,k.

Now we prove that + En Eu + EEji + Ekiki ++ En Eu Eki + En) A(Eij + EEikik + + En Eu + + Eji Eji + Eki Eu) --=- EE^ {j + EE ikik + j{ + and i + Eki + Eu Eji +En En ++Eji Eji ++Eki Eki ++Eu) Eu) = +EE^ A(En+E + EEik :+En + E A(Eu +i3Eij i ++ EEik == EE« ik ;+ i{ i + tj f+ ik+En j{ + Eki + EK for distinct i, j , fc, /. We consider the case i = 1, j = 2, k = 3, / = 4 as an example. For simplicity of writing we write down only the 4 x 4 submatrix

Chapter 6. Geometry of Hermitian

336

Matrices

situated at the upper left corner of the n x n matrix under consideration when all the other entries of the matrix are O's. By (6.29) / xu 1 1

/ 0 1 1 1 \ 1 1 \ 1

1 1 1 \

V l

where x l x = 0 or 1. Assume m = 1. The arithmetic distance between / 0 1 1 1 \ 1 1

M

/ 0 and

/

V

\ 1 1 1 1 1 /

is 3, but the arithmetic distance between (l

l

l

l

/0

\

i i

and

v i

V

/

\ 1 1 1 1 l /

is 2, which is a contradiction. Therefore /0 1 1 1 \ 1 1

/ o 1.1

Vi

Vi

/ 1 1 1 1 \ 1 1

/1 1 1 1 \ 1 1

Vi

Vi

I

\

l I

and consequently

/

/

The above argument can be used to prove inductively that for any s, where 1 < s < n, any 1 < ii < i 2 < • • • < z5 < n, and any j (1 < j < s),

-*(£(£.>* + Eiki,)) = 52(Eiiih + Eiki]) fc=l *±3

k=l k*3

6.4. Proof of the Fundamental Theorem (the Case n>3)

337

and

A®* + i2(Eiiik + Eiki})) = Ei}i] + ±(Ei]ik + Eikij). In particular, (6.35) holds also for the case F = F 2 . As in Case (a) we can prove in the same way that a = cr2, and ~xa — x~° for all x 6 D. After subjecting A to the bijective map Z i—► Zc~ , we also have (6.15) - (6.17), (6.22), (6.32) - (6.34), etc., and (6.35). (iv) As in Step (iv) of the proof of Theorem 5.4 for the case F being of characteristic not two in Section 5.4 we can prove by induction on the rank of the diagonal matrix [Ai, A2, • • •, An] that A([XU A2, • • •, An]) = [Ai, A2, • • •, An] for all Xu A2, • • •, An € F.

(6.41)

The case of rank 1 follows immediately from (6.35). Assuming that (6.41) is true for diagonal matrices of rank r — 1, where 2 < r < n, we are going to prove that it is also true for diagonal matrices of rank r. We consider the case when Ai, A2, • • •, Ar are all nonzero and A r + i = Ar+2 = • • • = An = 0 as an example. The other cases can be treated in a similar way. Thus we want to show that ^([Ai,A 2 > ---,A r ,0,--.,0]) = [A 1 ,A 2 > -..,A r ,0,...,0]

(6.42)

for all Ai,A 2 ,---,A r G F*. At first, in the same way as in Section 5.4 we can show that ^([A1,A2,...,Ar,0,.--,0])=

X)

x E

ij &

i<«,i
where x*-{ = x*- (1 < i,j < r). Then when r > 2, (6.42) can be proved in the same way as Section 5.4. But the case when r = 2 needs some modification. We write down the proof for the case r = 2 below. To simplify our writing we write down only the 2 x 2 submatrix situated at the upper left corner of the n x n matrix under consideration when all the other entries are O's. Thus 1

1 = 1 x^li x^12

A2 I

\ X\2 12

x2 2

22

338

Chapter 6. Geometry of Hermitian

Matrices

Since XiEu + X2E22 is adjacent with both XiEn and X2E22, we have rank ( " ' V ^ \

#12

X 2

f*12 . ) = 1,

\ )=iaak(&

x

22 /

\

X

x

12

22

—

(6.43)

^2 /

from which we deduce \lX*22 = X2x*n.

(6.44)

Since XiEu + A2J522 is adjacent with both AiJSu — A2(i?ii + Ei2 + E2i) and A22?22 - Xi(Ei2 + E2i + E22) which belong to C\{F) and C2{F), respectively, and are left fixed by A according to (6.35), we have r a n k f ^ i 1 ^ 2 y xj 2 + A2

* » + A* )= rank ( - ^ \ > + * \ U 1. z;2 ) \ x*12 + A2 z ^ - A2 + Ai / (6.45) From (6.43) and (6.45) we deduce Ai = Xn — X12 — #i2>

^2

=

#22 ~~ #12 ~~ #12*

(6.4o)

Substituting (6.46) into (6.44), we get [x12 + x12)x22 = (# 12 + ^12)^11-

(6.47)

Consider first the case Ai ^ A2. From (6.44) we deduce x^ ^ x22, then from (6.47) we deduce x\2 + x\2 = 0 and from (6.46) we deduce Ai = x\ly X2 = x22. Thus (X, \ _ ( X1 x*12\ [ A2 ; - {1% \2) ■ Since Ai^Bn + X2E22 and Ai.Sn are adjacent, we have

1

'Hit)' ' from which we deduce x\2 = 0. Hence -4([Ai,A3]) = [Ai,A2] f o r A ^ A j . Then consider the case \x = A2. Let A = Ai = A2, then A ^ 0. By (44) we have xjj = x22. Thus A(

X

\ _ ( xn

*ia ^

6.4. Proof of the Fundamental Theorem (the Case n > 3)

339

Assume that x^ ^ A. Since X(En + E22) and x^En + XE22 are adjacent, rank

( ^ ~ x*1- X I \

x

x

12

A

ll

=

lj

/

from which we deduce x{2 = 0. Then from the adjacency of X(En + E22) and XE11, and x^ ^ A, we deduce x\x = 0, which is a contradiction. Therefore xjj = A, and A

J

\ X19

A

Since A(.En + £22) and A2?n are adjacent, 0

JCJ2

Xc12 i9

A

rank [ - ^ - ~ » | = 1, from which we deduce x*2 = 0. Therefore >t([A, A]) = [A, A]. (v) Proceeding in the same way as Step (v) of the proof of Theorem 5.4 for the case F being of characteristic not two in Section 5.4 we can prove that any r (1 < r < n) and 1 < ii < i2 < • • • < ir < rc, •4( YJ

X

isitEisit)

=

l<s,t
X

*isitEi*it,

YJ l<s,t
where xitis = x~~ and x$tia = x~f~(l < s,t < r), for all z i a t s e F (1 < s < r) and ztst-t G JD (1 < s < t < r). (vi) We prove that for any r (1 < r < n) and 1 < z'i < %2 < • • • < ir < rc,

( 6 - 4 8)

.4( ^ *,-.A-,)= £ *<.A., l<s,t
l<5,t
where x tt , s = x ^ " (1 < s,t < r), for all X{aia e F (1 < s < r) and X{sit G i ) ( l < < s < ^ < r ) b y induction on r. Let r = n, then we obtain Theorem 6.4. When r = 1, (6.48) follows from (6.35). Now let us consider the case r = 2. We take the case i\ = 1 and z*2 = 2 as an example, i.e., we are going to prove that A

( *n

*i2 \

V Zl2 #22 /

( xii

x12 \

V ^12 ^22 /

for

a H

^^^ ^

a n d

^

^ ^

4 9

Chapter 6. Geometry of Hermitian

340

Matrices

When si2 = 0, (6.49) follows from Step (iv). When xu = x22 = 0, (6.49) follows from (6.32) and (6.33). Now consider the case when x22 = 0 and X12 7^ 0. By (6.17) we have x\2 = 0, i.e., A

(

*11

*12

( fli

"\ _

\xi2

o ;

V ^

*12

\

o ;*

Since X n £ n + ^12^12 + ^12^21 and x12E12 + x^E2i are adjacent, we must have For any x £ F the arithmetic distance between x n ^ n + ^12^12 + ^12^21 and ZiiiJn + ^^22 is 2, therefore rank ( ^ i j j * 1 1 \ Z12

*12 ) = 2 for all x G F. -x J

It follows that xn = xu. Hence (6.49) also holds for the case £22 = 0 and x12 7^ 0. Similarly, (6.49) also holds for the case xn = 0 and xu ^ 0. Finally let us consider the case when all x u , xi 2 , #22 are nonzero. For any x the arithmetic distance between x^Eu + x12E12 + x~^E2i + #22^22 and #11^11 + xE22 is 2, therefore rank ( \

I

»-^ri1 x

\2

*» x

22

~~ x

) = 2. )

It follows that xu = Xu. Similarly, x\\ — x22- Since x\\E\\ + Xi2Ei2 + #iii?2i + 2:22^22 and #n.Eii + x\2E\2 + #12^21 are adjacent, we must have #12 = ^12- Therefore (6.49) also holds for this case. Now assume that (6.48) is true for r — 1 (3 < r < rc), we are going to prove that it is also true for r. Take the case z'i = 1 , ^ = 2, • • •, ir = r as an example. We may proceed as in Steps (vii) - (x) of the proof of Theorem 5.4 in Section 5.4 with some modifications and will arrive at the conclusion that A( J2

x

ijEij)=

Y,

x

ijEij,

where xj{ = x~J (1 < ij < r), for all xi{ G F (1 < i < r) and Xij € D (1 < i < j < r). One modification is to replace the determinant argument used there by the rank argument which works also for division rings as we

6.5. Maximal Sets of Rank 2 (the Case n = 2)

341

did in Steps (iv) and (vi) above. Another modification is an argument in Step (viii) of the proof of Theorem 5.4 in Section 5.4. Let A(

Y, l
X

iJEi3

+ Z r - l , r £ r - l , r + X~^Er^r

+ XrrErr)

=

£) X*jEijy l
where Xji = x~Tj (1 < i,j < r — 1) and x*- = ~xfj (1 < i, j < r). As in Section 5.4 we prove that x*- = Xij (1 < i < r — 1,1 < j < r — 2), x*T = 0 (1 < i < r — 2), x*_lr = x r _ l j r , z*r = xrr. But the proof of x*_lr_1 = x r _i >r _i should be modified as follows. We choose a \x G F for which p, ^ xrr. For such a ^ there is a unique X E F such that rank f ^ i ^ i r y

A

«^r—l,r

Xr

~^

»^rr

) = 1. ^

/

Then /J

^o-E't-j + x r _i >r £' r _i, r + £r_i>rJEr>r_i +

xrrErr

l
is adjacent with 53

ar,jJ5y + (A - Sr-i^-iji?,.-!^.! + /z£ r r .

l
According to Step (vii) the latter is left fixed by A. It follows that rank ( X*-^~1 " ^ z r -i,r

A

Xr r '^ ) = 1, xrr - p, J

and, hence, x*_ l j r - 1 = £r_1)7._i. The inductive proof is now completed.

6.5

□

Maximal Sets of Rank 2 (the Case n = 2)

In this section and the next we assume that n = 2 and Z) is a field which possesses an automorphism — of order 2. Let F be the fixed field of —. It is well-known (cf. Jacobson 1974) that D is a quadratic extension of F. When D is of characteristic not two, there is an element p G D\F such that ~p = —p and D = F{p). When D is of characteristic two, there is an element p G D\F such that ~p = p-\-1 and D = F(p). Note that Assumptions 1° and 2° stated in Section 1 are now automatically satisfied.

342

Chapter 6. Geometry of Hermitian

Matrices

Let H2(D) be the space of 2 x 2 hermitian matrices over D. We have Mi = {tuEu |fo G F } , d = {xnEn + x12E12 + x5E2i

t = l,2,

\ xi{ G F, x12 G £>},

* = 1,2

(6.50)

and we know that Mi and JM 2 are maximal sets of rank 1 (cf. Proposition 6.8) and that C\ and C2 are the unique maximal sets of rank 2 containing Mi and M2, respectively, (cf. Corollary 6.16). We have also (6.5) MX2 = {£(£n + E12 + E21 + E22) | £ G F } . We know that M\2 is also a maximal set of rank 1, that M\C\ M2C\ M\2 — {0}, and that any three distinct maximal sets of rank 1 which have a nonempty intersection can be carried simultaneously under a transforma­ tion of the form (6.1) to M\, M2, and M\2, respectively, (cf. Proposition 6.13). Proposition 6.28: Let £

_ l I

#11 Zll+^12

Xu + X12 ^11 + ^12+^12

* n eF,

x12eD\.

Then C\2 is the unique maximal set of rank 2 containing

(6.51)

M\2.

Proof: We have

1

Ik

1

Jl-A.,,

and

l

\YV\

h i

; I = C12.

Hence our proposition follows from Corollary 6.16.

□

Let us study the intersection of maximal sets of rank 2. From (6.50) and (6.51) we deduce Proposition 6.29: The intersection of C\ and C2 is

An£2=

{(4

T)|*»€4

(6 52)

-

6.5.

Maximal Sets of Rank 2 (the Case n = 2)

343

the intersection of £ i and £ i 2 is

; £ An£n 12 = {(^_L

' M ( ™ ?)

*u 6 D L ,

(6.53)

and the intersection of £2 and £12 is ( ££ 22 n£ H £12 = j 12 = \

0

x12

«i2

\

z 1 2 + X12 /

4

x 1 2 € Z) I .

(6.54)

Moreover, when D is of characteristic not two,

; - ) z€Fj, -{(::)!"'}■

d nnr£22 nn,£112 A 2=

(6.55)

and when Z) is of characteristic two

An£ £1 n £2 n£ n £12 12 = {( y ° j ) ye A

(6.56)

2

□ n

Now we study the intersection of three distinct arbitrary maximal sets of rank 2 which have a nonempty intersection. By Proposition 6.15 we can assume that one of them is C\. Let the other two be denoted by £ ' and £". Suppose that H0 G £ i n £ ' n £ " . Then the transformation Z —► Z—H0 leaves £1 fixed. Denote the images of £ ' and £ " under this transformation again by £ ' and £", respectively. By Proposition 6.18 there is a unique maximal set of rank 1 contained in £ ' and containing 0, which will be denoted by M!. Similarly, there is a unique maximal set of rank 1, denoted by M!{\ contained in £ " and containing 0. By the proof of Proposition 6.12 we can assume that

(i.

)

M2(

A

0

-1

5

where X E D. Then by the proof of Proposit ion 6.13 we can assume that t

M" =

(IT rr

)

-1

M12 ('

ft,

where h £ D* • B y Proposit ion 6.18 we have

'IT

1 A

0

1 (i

,r

5

Chapter 6. Geometry of Hermitian

344 and

t-

i'

£"=

|

N\

1

<

A

* I-

—

\ , *J H u o-

' 1

\ .

1 ,) ( ,

*

Clearly, we also have t

£' =

■"(TTIT"

By (6.54), we have

Matrices

/' 1

/ 1

\

.w *) U0-

a n C"

iyi w i MA

lj

wo

\ ( \hx12 + \hx12 + Whh(xi2 ~ \ \

^ 1 2 + \hh(xi2

ere XAO £ D.

*ia wi wi

h ) \xT2~ x12 + xfi J \

\

+ x~^2~)

h) \ X 1

+ £12)

hh(x12 + ^12)

Therefore

\hx12 + \hx\2 A n £ n c" = | ( hxi2

hx\2 0

I

hxi2 + Xhh(xi2 + x~^)

#12 £ D and x\2 + #12

-}■

When Z) is of characteristic not two, we have #12 = pz, where z G F. Let ft = c + p~1c and Xhp + Xhp = a, then a, 6, c G F and (6, c) ^ (0,0). Thus we have

^*"*"*-{{*?*, ' " ^ { ( f c ; ; k bz+ r r) )>tf}, H4 where a,b,c e F and (6, c) ^ (0,0). When Z) is of characteristic two, we have #12 = y G F . Let h = b+ pc and \h + \h = a, then a,i>,cG F and (6,c) ^ (0,0). Thus we have

'n£' = clnc'ncr =U l(. [\by where a,b,c€

ay

hy pcy

_

+ pcy

0

+- )yeFver], )\ }' J

F and (6,c) ^ (0,0). We have proved

Proposition 6.30: Let £ i , £', £ " be three distinct maximal sets of rank 2 and £ i n £ ' D £ " ^ <£. Then

a hbz+/cz )+H \Z6F\,'€F\, £^xn£'n£" = (f. J_ ^■{(k+% r ) + * Q

[y oz + pcz

0

6.5. Maximal Sets of Rank 2 (the Case n = 2)

345

where a, 6, c G F and (6, c) ^ (0,0), and H0 G A H £ ' fl £".

□

Proposition 6.30 suggests the following definition. Definition 6.9: Let £ be a maximal set of rank 2. A line in C is either a maximal set of rank 1 contained in C or the intersection of C with another two distinct maximal sets of rank 2 when it is nonempty. □ From Proposition 6.30 and Definition 6.9 we deduce Proposition 6.31: Let £ be a maximal set of rank 2, then C has a three dimensional affine space structure over F with the lines defined in Definition 6.9. More precisely, let C be in the following general form

H

x

Kyy

+ pz pz +

y + pz )P + H0 x,y,z G F l , 0

where P G GL2(D) and H0 G H2(D), then the map

c —^ --♦ C

tPi v

x y + pz *)P + H 0 y + ~pz 0

--»

is a bijective map carrying lines in C into lines in

AG{3,F) AG{3,F)

(x,y,z) AG(3,F).

□

When D is of characteristic not two, define

£ w

- ={(;

0

0 pz ~pz w J

^={U

x,zeF>,

(6.57)

z,w G F\.

(6.58)

/

When D is of characteristic two, define

A(*T = { ( :

2)

x,yeF\,

(6.59)

y,w e f l .

(6.60)

and

A(F)'=| Then we have

C

346

Chapter 6. Geometry of Hermitian

Matrices

Proposition 6.32: Let D be of any characteristic, then

c1(Fy = (c1nc2nc12) u{x eC1\3YeC1nc2n

c12j such that ad(x,r) = 1}

(6.61)

U{X G £21 3 F G & n C2 n £ 1 2 , such that ad(X, r ) = l } .

(6.62)

and

c2(F)f = (c1nc2nc12) Proof: We prove only (6.61) in the following, since (6.62) can be proved in a similar way. We consider only the case when D is of characteristic not two, the other case can be proved in the same way. By (6.55) and (6.57) it is clear that £i(f)'c(£in/;2n£12) U{X G £ i | 3Y e £ i nC2C)C12,

such that ad(X, Y) = 1}.

Conversely, it is also clear that £in£2n£i2c£i(F)'. Suppose that X G £ i and that there exists Y G C\ f) C2 fl £ i 2 such that a d ( X , y ) = 1. Let _ / xlx x12 \ X

-[xT2

0 )

and

Y- ( ° ^ ^ V ?* ° ) ' where z <E F. From a d ( X , F ) = 1 we deduce a:12 = /02 and xn Therefore X € A ^ ) ' - Thus

^ 0.

{X € £11 3 F € A n £ 2 n £ 1 2 such that ad(A", F ) = 1} c - d ^ ) ' Hence ( A n £ 2 n £ 1 2 ) U {X € A | 3 F G £ i fl £2 (1 £ 1 2 , such that ad(X,Y) C £x(F)'.

= 1}

6.5. Maximal Sets of Rank 2 (the Case n = 2)

347

(6.61) is now proved.

□

Parallel to Proposition 6.30 we can prove Proposition 6.33: Let C be a maximal set of rank 2 in H2(D) and assume that C\ ^ C and \C\(F)' fl C'\ > 2. When D is of characteristic not two, then C^F)' H C is of the form

+ i€F ^^ F))'' nn££ ''={(;' T ) = { ( ^ T ) + H^ } X€F\,

where a G F, c G F*, and iJ 0 € Ci(F)' H C. When D is of characteristic two, then Ci(F)' fl C is of the form

1 ax bx \ bx 0

Ci(F)'nc' = l where ae F,be

)+#o

X€F\,

F*, and # 0 G A ( ^ ) ' n C.

□

Parallel to Definition 6.9 we have Definition 6.10: Let C^F)' be the set defined by (6.57) or (6.59) when D is characteristic not two or two, respectively. Then a line in C\(F)' is either a maximal set of rank 1 contained in Ci(F)f or the intersection of Ci(F)' with any maximal set of rank 2 of ^(D) when it contains at least two points. □ From Proposition 6.33 and Definition 6.10 we deduce Proposition 6.34: When D is of characteristic not two, the map A(F)' A(F)' ( x

\pz

pz\

0 j

—♦ —♦

A AG(2,F) G(2,JF)

I—►

(z,z)

is a bijective map which carries lines in £ i ( F ) ' to lines in AG(2,F). D is of characteristic two, the map A (F)' d(Fy

When

—► AG(2,F) AG(2,F) i—►

(x,y)

is a bijective map which carries lines in C\(F)' to lines in AG(2, F).

□

Chapter 6. Geometry of Hermitian

348

Matrices

Parallel to Propositions 6.33, 6.34 and Definition 6.10 we have Proposition 6.35: Let £ be a maximal set of rank 2 in H2(D) and assume that C2 ^ C and \C2{F)' n C'\ > 2. When D is of characteristic not two, then C2(F)' 0 C is of the form

«^-{(;s)** s e f j , where c G F*y d G F , and H0 G £ 2 ^ ) ' H £'. When J9 is of characteristic two, then C2(F)' fl £ ' is of the form

/wrntf-{(£ ^)+«, » € P | , where beF\deF,

and # 0 € £ 2 ( F ) / 0 £'.

□

Definition 6.11: Let £ 2 ^ ) ' be the set defined by (6.58) or (6.60) when D is characteristic not two or two, respectively. Then a line in C2{F)' is either a maximal set of rank 1 contained in C2{F)' or the intersection of C2(F)' with any maximal set of rank 2 of H2(D) when it contains at least two points. D Proposition 6.36: When D is of characteristic not two, the map

C22(F)' (F)' —> —♦ AG(2,F) AG(2,F) ( 00 pzpz\ 1 ► (Z,W) v ' \ pz pz w w J' is a bijective map which carries lines in C2(F)' to lines in AG(2,F). D is of characteristic two, the map

When

d(F)' —► AG(2,F) AG(2,F) C[FY —♦ 0 y (0 a i—► (w, ( ? / , to) w)

\y

wJ

is a bijective map which carries lines in Ci{F)' to lines in AG(2, F).

6.6

□

Proof of the Fundamental Theorem (the Case n = 2)

We prove only the first statement of the theorem. We proceed in steps.

6.6. Proof of the Fundamental Theorem (the Case n = 2)

349

(i) Let A be a bijective map of H2(D) to itself such that both A and A'1 preserve the adjacency. As in the case when n > 3 we assume that (6.15) A(0) = 0. By Proposition 6.13 we can assume that (6.16) A(Mi)

= Mu

* = 1,2

and A{MU)

= M12.

(6.63)

By Corollary 6.16 we have (6.17) A(d)

= Cu

i = 1,2,

and A(C12) = £ 1 2 .

(6.64)

A(& fi C2) = A f l £ 2 ,

(6.65)

A(CX n £i 2 ) = £1 n £12,

(6.66)

A(C2C\£12) = C2nC12,

(6.67)

A(CX n £ 2 n £12) = £1 n £ 2 n C12.

(6.68)

Thus

and It follows from Proposition 6.32 that A(Ci(F)')

= Ci(F)',i

= 1,2.

(6.69)

(ii) We are going to show that after subjecting A to a bijective map of the form (6.2), we can assume that besides (6.16) — (6.17) and (6.63) — (6.69), we have also A{X) = X for all X G Ci{F)\ i G 1,2. (6.70) Consider first the case when D is of characteristic not two. By (6.69), for any x, z G F we have A

A

(

\pz

x

Pz \ - (

0 )-{pz*

x

*

Pz*\

0 j '

350

Chapter 6. Geometry of Hermitian

Matrices

where x*, z* G F. By Proposition 6.34, the fundamental theorem of the affine geometry, and *4(0) = 0, we have (x* (x*,z') ,*•) === (a;,(x,zYP,

.*rp,

where a is an automorphism of F and P G GL 2 (F). From A(Mi) = M\ and A(Ci D £2 H £12) = £1 fl/^2 n £ i 2 we deduce that P is a diagonal matrix. Let P = [pi,P2]5 where pi,p2 € F*. We may extend a to an automorphism of D, denoted by cre, as follows aee ::
D —> —► D D x + PV '—> 1—> x a + ^

(a;,i/G (a;,?/G F ) .

Clearly, cre is an automorphism of D commutative with the involution —. After subjecting A to the bijective map 1

1 1

1I

X H*—+ - + ( p 1 * (pxtp^XPp - x p - ) *)'* « 9,

we can assume that A{X) = X for f oall r a lXl XGG££ii(:(Ff )) '' . ■ * ( * ) == X

(6.71)

Of course we also have (6.15) - (6.17) and (6.63) - (6.69). By (6.16) we can assume that

42

0 0 \ X

/

0 0 ■I f°

10

where x,x* G F*. Since

/ /vw px \ 00 ) V px PX ;

and

V)

y 0 x x ) \0

are adjacent, we have x* = #, i.e., , I' 0 A A\

fl\

/o 0 \

{1\.° */ x ) == (S v°n *«J)

for allfora11 x € F.

Similarly, by (6.69), for any z,w G i71 we have pz*\ .4 C' 0_° pz^ \ )_-=( -(_° '** •"( • \ ^ pz

/>* w* y ' ww )J ' VI />z*

*^

(6-72)

6.6. Proof of the Fundamental Theorem (the Case n = 2)

351

where z*,w* G F. By Proposition 6.36, the fundamental theorem of the affine geometry, and *4(0) = 0, we have TT

{z*,w*) Q, »w*) ==--{z.(z,w) w) Q, (** where r is an automorphism of F and Q G GL2(F). we deduce Q = I and r = 1. Therefore we have

From (6.71) and (6.72)

A(X)-X G A(X) ==Xx for all X €C A2((F)'. F)'.

(6.73)

When D is of characteristic two, though the forms of £ i ( F ) ' , i = 1,2, are different from the characteristic not two case, but using the above argument we can also achieve the conclusions (6.70). (iii) Now we are going to show that after subjecting A to bijective map of the form (6.2), we can assume that besides (6.15) — (6.17) and (6.63) — (6.70), we also have

,4(Z) A(Z) == ZZ for all Z € G £,, A , i* = = = 1,2.

(6.74)

At first we assume that F ^ F 2 . By (6.17) we have x A [^y + pz W \

y + pz

yy* ++ pz* y + pz\ pz^ _ ( x* 0 , Vy*+^* 0 0

J

= ( . * * . * /**lJ \y* + pz*

0

where x,y,z,x*,y*,z* G F. By Proposition 6.31, the fundamental theorem of the affine geometry, and (6.15), we can assume that (or*,y* ,*•) == (x y,

*ri/>,

where cri is an automorphism of F and P G GL3(F). following two cases.

We distinguish the

(a) D is of characteristic not two. From (6.70) and (6.65) we deduce that P is of the form 0 \ / i1 0

pP ==

0 pP22 P23 P23 22

X

Vo

0o

I/

22

•

1i ;/

Then

'22 11)\ •4( 0 ) * \ ,11 oj

_

p22 ++ ^~PP23 \I P22 23

p22 ++ /0P23 PP23 \ ^22 00

) '

352

Chapter 6. Geometry of Hermitian

Matrices

Since 2Eu + E12 + E21 G A D C12 and (6.66) holds, ( 2 P22 + PP23 PP23 ^ \ G A n£i2. />P23 V #22 P22 + ~PP23 0 }

We have 2 = P22 + /5P23+P22 + PP23 = 2p22, since p+~p = 0. Therefore p22 = 1. 2E1X+E12 + E21 and -\E22 are adjacent, and by (6.72) A{-\E22) = -\E22. It follows that p23 = 0. Therefore P = I. From (6.71) we deduce <7i = 1. Hence A{Z) A .■ .A(Z) == Z for all Z G6 C\

(6.75)

A(Z) --= = Z for all all Ze Z € C£22.-

(6.76)

=z

Similarly,

(b) D is of characteristic two. From (6.70) and (6.65) we deduce that P is of the form / l1 0 00 \ 0 . 0 1 ^P== P33 \ 0 p32 2 #33 ) 3 Then A ( 1 A \~P v? ' ' \

P

l1 P32 + AP33 \ \ =_ (( PS2+PP33 ^ = V P32 + PP33 ^33 ) ' 0 0) / 0

Since Eu + pE12 + pE21 G Cx n £12 and (6.66) holds, (

P32 + PP33 1 /?P33 ^ \ G£l(l£i2. P32 + ^ 3 3 0 \ P32 ~PP33 J

We have 1 = p32 + pp33 + p32 + ~pp33 = p 3 3 , since p + p = 1. E n + / ^ + pE21 and /9^E22 are adjacent and by (6.72) A(p'p) = p~p. It follows that p23 = 0 or 1. If p23 = 0, then P = / and <7i = 1. Hence (6.75) holds. Now let p23 = 1, then y + pz y + pz ) ■ ■ ( y + pz 0 y \ y + /&* 0 + pz )• ' ( After subjecting A to the bijective map

z-

-> Z for all Z G€ H U2(D) Z—>~Z we have (6.75). Similarly, we also have (6.76).

6.6. Proof of the Fundamental Theorem (the Case n = 2)

353

There remains to consider the case F = F2. Then D = F22 = W2(p), where ~p — p + 1 and pp = 1. By Step (i) we can assume that -4(0) = 0, A(Eu) = En (i = 1,2), A{E12 + E21) = E12 + E21, and A{Eu + E12 + E21) = £ « + £12 + £21 (» = 1,2). Let

V 0 1 ; - v x s *22 y ' where i n , £22 € -f and £12 G JD. Since JSn + E22 is adjacent with both £11 and £22, we have xn = x22. Suppose that Xn = 0, then from the adjacency of En + E22 and E\2 + E2U and A(E12 + £21) = E12 + E2\, we deduce rank ((x12 — l)Ei2 + (x~^— l)E2i) = 1, which is impossible. Therefore £11 = 1. From the adjacency of Eu + E22 and En we deduce xi2 = 0. Hence A(En + E22) = En + E22. Let / *ii

( 1

1 \

VI

I )~{x^

S12 \

x22 ) '

where i n , x22 G F and xi 2 G J5. Since En + Ei2 + E2i + E22 is adjacent with both En + £12 + E2i and £"12 + £21 + E22l we have i n = x22. Suppose that Xn = 0, then from the adjacency of En-\-Ei2 + E2i-\-E22 and En + Ei2 + E2i, we deduce x12 = 1, which contradicts A(Ei2 + £21) = £12 + E2i. Therefore Xn = 1. Then from the adjacency of En-\-E\2+E2\+E22 and En+Ei2+E2i we deduce £12 = 1. Hence A(En + #12 + E2i + E22) = En + £12 + ^21 + E22. We conclude that we can assume that A{X) = X

forallXG<S 2 (F 2 ),

where <S(F2) is the set of 2 x 2 symmetric matrices over F 2 . Consider the image of pEi2 + ~pE2\ under A. Since pE\2 + ~pE2\ G C\ C\ £2, by (6.65) we have

^;!)-(;o-(;s)Similarly,

^s)-(;o-(:o-

Chapter 6. Geometry of Hermitian

354

Matrices

There are two possibilities. Suppose that A(pEi2+~pE2\) = ~pE12 + pE2i and A(~pEi2 + pE2i) = pEu+~pE2i. Then from the adjacency of pEu + 'pE^ and En + pE12 + pE21 we deduce A(En + pE12 + pE21) = En + pE12 + pE21. Similarly, A{En + pE12 + pE21) = En+pE12 + pE2U A{pE12 + pE2i-\-E22) = pE12 + pE2i + £ 2 2 , A{pEl2 + pE2i + £22) = PE12 + pE2i + E22. Hence we conclude that we can assume that both (6.75) and (6.76) hold. Now suppose that A(pE12 + pE2i) = pE12 + pE2i and A{pE12 + pE2i) = pE12 + ?J5 2 i. After subjecting A to the bijective map Z—>Z, we reduce the present case to the above case. (iv) We can prove that A([Xi, Ax]) = [Ax, Ai] for all Ai, A2 G F by the same argument in Step (iv) of the proof of Theorem 6.4 when n > 3 in Section 6.4. (v) Finally, we can prove that A{X) = X for all X

eH2(D)

by the same argument in Step (vi) of the proof of Theorem 6.4 when n > 3 in Section 6.4. The proof of Theorem 6.4 for the case n = 2 is now completed.

□.

Without essential difficulty the proof of Theorem 6.4 for the case n > 3 and the case n = 2 can be generalized to prove the following Theorem 6.37: Let D and D' be division rings which possess involutions — and ~ , respectively, and assume that both (D,—) and (D\~) satisfy Assumptions 1° and 2°. Let n and n' be integers > 2. When n = 2, we assume further that D is a field, and when n' = 2, we assume further that D' is a field. Let A : 7in(D) —4 Hn>(D') be a bijective map such that both A and A~l preserve the adjacency, then n = n', D and D' are isomorphic, and A is of the form A(Z) = a'PZ^P

+ Ho for all Z 6 Hn(D),

(6.77)

6.7. Application to Algebra

355

where a G JF'*, P G GLn(D'), H0 G Hn(Df), and cr is an isomorphism from D to D' for which aa = (a a )~ for all a £ D. Conversely, if D is isomorphic to D', then any map of the form (6.77) from Hn(D) to Hn'(D') is bijective, and both the map and its inverse preserve the adjacency. □

6.7

Application to Algebra

In this section we assume that D is a division ring of characteristic not two, which possesses an involution —. Clearly, the set Hn(D) of n x n hermitian matrices over D forms a Jordan ring with respect to the addition

(A, B)^-*A

+B

and the symmetrized multiplication

(A, B)^^(AB

+ BA).

This Jordan ring is called the Jordan ring of hermitian matrices and is denoted by J(7in(D)). Parallel to Section 5.8 we can show that the following theorem holds. Theorem 6.38: Let D and D' be division rings which possess involutions — and ~ , respectively, and assume that both (Z), —) and (D\~) satisfy Assumptions 1° and 2°. Let n and n' be integers > 2. When n — 2, we assume further that D is a field, and when n' = 2, we assume further that D' is a field. If there is an isomorphism of Jordan rings A : J{Hn{D))

—>

J(Hnl(D')),

then D is isomorphic to D', and A is of the form A{Z) = at~PZaP,

(6.78)

where a G F'*, P G GLn(D') and satisfies *PP = a - 1 / , and a is an isomor­ phism from D to D' for which ~a? = (a*7)" for all a G D. Conversely, the map of the form (6.78) from J(Hn(D)) to J^^D')) is an isomorphism of Jordan rings. D

Chapter 6. Geometry of Hermitian

356

6.8

Matrices

Application to Geometry

In this section we assume again that D is a division ring of any charac­ teristic, which possesses an involution — and n > 2. We assume also that both Assumption 1° and 2° hold. The discussion in the present section is parallel to Sections 4.4 and 5.9, so we state the definitions, propositions, and theorems, and give only the different proofs. Let

K-( K

/(n)

°

- { _/
0

A 2n x 2n matrix T over D is called a unitary matrix with respect to K if l

TKT

= K.

Clearly, 2n x 2n unitary matrices over D with respect to K are invertible and they form a group with respect to the matrix multiplication, called the uni­ tary group of degree 2n over D with respect to K and denoted by U2n(D^ K). It is easy to check that the following matrices belong to J7 2 n(^ ? K): 1°

l

A

t/^ix)i

1

where A € GL n (£>).

H

j j , where <# = H.

T_ r

r

I 5 where J 2 = J is a diagonal matrix.

Let m be an integer, 0 < m < n, and P be an m-dimensional subspace of the 2n-dimensional row vector space D^2n\ Define

P± =

{v£D^\vKtT=Q}.

Clearly, PL is a subspace of D^2n\ and we call PL the dual subspace of the subspace P relative to K. Clearly, we have PLKtP = 0 and by Theorem 1 1.27 d i m P = 2n — m. It is easy to see that the map P ^ P

1

6.8. Application to Geometry

357

from the set of subspaces of Z)(2n) to itself is bijective and has the following properties: for any two subspaces P and Q,

D QL, pP ccQQ => ^PLPL =>Q\ and for any subspace P , 1 d i m P - f ddi imr nPP- 1 = 2n,

and (p±)± {PL)L ■== pp. If we regard an m-dimensional subspace P as an (m — l)-flat in PGl(2n — 1,D), then P x is an (2n — m — l)-flat, called the dual flat of the flat P . Therefore the map P —> P - 1 from the set of flats in PGl(2n — 1, Z)) to itself is a polarity of PGl(2n — 1,1?), called the unitary polarity defined by the 2n x 2n invertible skew-hermitian matrix K. If P 1 = P , then P is called self-dual and the corresponding subspace P is also called self-dual. Clearly, self-dual subspaces are of dimension n and (1^ 0^) is a self-dual subspace. Parallel to Proposition 4.18 we have Proposition 6.39: A subspace W is self-dual if and only if dim W = n and WK*W

= 0.

a

An m-dimensional subspace P is called totally isotropic (with respect to K) if PK *P = 0. A totally isotropic subspace is called maximal totally isotropic if its dimension is equal to the maximum of the dimensions of totally isotropic subspaces. By Witt's Theorem maximal totally isotropic subspaces are of dimension n. Parallel to Propositions 4.19 and 4.20 we have the following propositions. Proposition 6.40: A subspace is self-dual if and only if it is a maximal totally isotropic subspace. O Proposition 6.41: Let W be an n-dimensional subspace. Write a matrix representation W of the subspace W in the block form

w =-- (x n

(6.79)

Chapter 6. Geometry of Hermitian

358

Matrices

where both X and Y are n x n matrices. Then W is self-dual if and only if XtY

= YtY.

(6.80)

□ Let W be a self-dual subspace. If in the block form (6.79) of W r a n k X = n, then W has a matrix representation (/(») x-xY).

(6.81)

From (6.80) we know that X~XY is an n x n hermitian matrix. Hence the set consisting of all self-dual subspaces is called the projective space ofnxn hermitian matrices over D and denoted by VHn{D). It is also called the dual polar space of type 2A2n-\. The self-dual subspaces are called the points of VHn(D). Any matrix representation (6.79) of a point W G V7in(D) is called a homogeneous coordinate of the point. If the matrix X in (6.79) is of rank < n, then the point W is called a point at infinity. If the X in (6.79) is of rank rc, then W is called a finite point, and it has (6.81) as one of its homogeneous coordinates and the n x n hermitian matrix X_1Y as its non-homogeneous coordinate. As in section 4.4, restricting the actions of GL2n(D) on Z)(2n) and on the set of subspaces of D<2n) to U2n(D,K), we obtain actions of U2n(D, K) on £)(2n) and on the set of subspaces of D^2n\ Parallel to Propositions 4.21 — 4.26, Corollary 4.27, and Definitions 4.6 — 4.7, we have Proposition 6.42: U2n(D,K)

acts transitively on VHn(D).

D

Definition 6.12: The arithmetic distance between two points W\ and W2 of VHn{D), denoted by ad(Wi, W2), is denned to be r, i£dim(W1+W2) = n+r. When r = 1, they are said to be adjacent. D Proposition 6.43: Let Wx and W2 be any two points of VHn(D).

Then

ad(Wi, W2) = rank WXK%W2. D

Proposition 6.44: Let Wu W2 and W3 be any three points of Then

VHn(D).

6.8. Application to Geometry

359

1° ad(Wi, W2) > 0; ad(WuW2)

= 0 if and only if Wt = W2.

2° ad(Wi, W2) = ad{W2, Wt). 3° ad(Wi, W2) + ad(W 2 , W3) > ad(Wi, W 3 ). D

Proposition 6.45: Let W\ and VF2 be two points of VHn(D) and as­ sume that ad (Wi,W2) = r. Then they can be carried simultaneously under U2n(D,K) to

(/ 0) and y

Q

/(n _ r)

Q

Q(n_r)

j. D

Proposition 6.46: U2n(D, K) leaves the arithmetic distance between any two points of VHn(D) invariant. Moreover, for any fixed r, 0 < r < n, the set of pairs of points of VHn(D)y which are of arithmetic distance r, forms an orbit under U2n{D,K). □ Definition 6.13: Let W, W G VHn(D). When W ^ W, they are said to be of distance r, denoted d(VF, W) = r, if r is the least positive integer for which there is a sequence of r + 1 points WQ = W, W\, W2, • • *, Wr = W such that W{ and Wi+i are adjacent, i = 0,1,2, • • •, r — 1. When W = W, we define d(W,W) = 0. □ Proposition 6.47: For any two points W, W G

VHn(D),

&(W,W') = *,&(W,W'). D

Corollary 6.48: Let A be a bijective map from V7in(D) to itself and assume that both A and A'1 preserve the adjacency of pairs of points in VHn(D), then A preserves also the arithmetic distance of any pair of points in VHn(D). □ We call a 2n x 2n matrix T over D to be generalized unitary with respect totfif *TKT = aK,

Chapter 6. Geometry of Hermitian

360

Matrices

where a G F*. The set of all 2n x 2n generalized unitary matrices over D with respect to K also forms a group with respect to the matrix multiplication, called the generalized unitary group of degree 2n over D with respect to K and denoted by GU2n(D,K). Every element T G GU2n(D,K) defines a bijective transformation on VHn(D)

W^->WT, which preserves the arithmetic distance between any pair of points. In par­ ticular, both the map and its inverse preserve the adjacency of pairs of points of VHn(D). Conversely, we have the fundamental theorem of the projective geometry of hermitian matrices, which reads as follows. Theorem 6.49: Let D be a division ring which possesses an involution — such that Assumptions 1° and 2° hold. Let n > 3 and when n — 2 assume that D is a field. Let A be a bijective map of V?{n(D) to itself and assume that both A and A~x preserve the adjacency of pairs of points. Then A is of the form W ►-* W'T

for all W G VHn(D),

(6.82)

where a is an automorphism of D which is commutative with the involution — and T G GU2n(D^K). Conversely, any map of the above form from VHn{D) to itself is bijective and both the map and its inverse preserve the adjacency of pairs of points of VHn{D). □ To prove this theorem we need the following preparations. Let W G VHn(D)

and write W = (w1w2

••• w2n),

where wu w2, • • • ,w2n are the 2n columns of W. Let s be an integer, 0 < s < n, and ii, i 2 , • • •, is,jij2, • • • Jn-S be a permutation of 1,2, • • •, n such that 1 < ij < i2 < • • • < is < n, 1 < jx < j 2 < • • • < j n _ s < n. Define ftw-is

= {W e VHn{D)\{wil

• • • wis wn+jl

• • • wn+jn_3) is invertible}.

When s = 0, we denote H = {W e VHn(D)\(wn+i

wn+2 • • • w2n) is invertible}.

6.8. Application to Geometry

361

Then parallel to Lemmas 4.29 and 4.30 we have the following two lemmas. L e m m a 6.50: For any W € Wtlt-2...ia, multiplying W from the left by the inverse of (—w^ — wi2 • • • — wia wn+jl wn+J2 • • • wn+jn_9), we obtain a matrix representation of W whose z'i-th, zVth, • • •, z s -th, ( n + j i ) - t h , (rc+J2)th, • • •, (n + jfn_5)-th columns form the matrix

J(ns) J

^

and the remaining n columns can be arranged into an n x n hermitian matrix over D z = {wn+il wn+i2 • • • wn+is wh wh • • • wjn_s).

(6.83)

Moreover, the map /

Hi1is...ia —> Hn(D) W ^ Z

is a bijection which preserves the arithmetic distance between pairs of points.

□ The n x n hermitian matrix Z in (6.83) is called the non-homogeneous co­ ordinate of the point W G Wtli2...t-5. L e m m a 6.51:

vnn{D) =

|J 1

<*l<*2<

nilis...ia. <*3
0<s
□ Parallel to Lemmas 4.32 and 5.52 we have L e m m a 6.52: Let n > 2, A be a bijective map from 7in(D) to itself, and assume that both A and A'1 preserve the adjacency of pairs of points of T-tn{D). Let io be a fixed index where 1 < z'o < n. Assume that A leaves every point of T-ln(D) whose (z0, z'o)-entry is nonzero fixed, then A leaves every points of Hn(D) fixed. □ This lemma can be proved in the same way as Lemma 5.52. After the foregoing preparations we can prove Theorem 6.49 in the same way as Theorem 5.49.

Chapter 6. Geometry of Hermitian

362

Matrices

Without essential difficulty Theorem 6.53 can be generalized as follows Theorem 6.53: Let D and D' be division rings which possess involutions — and ~ , respectively, such that Assumptions 1° and 2° are satisfied for both D and D', and let n and n' be integers > 2. When n = 2, assume that D is a field, and when n' = 2, assume that D' is a field. Let A be a bijective map from VHn(D) to VHn(D') and assume that both A and A'1 preserve the adjacency. Then n = ra', D is isomorphic to £)', and A is of the form W^-*WaT

for all W eVHn{D),

(6.84)

where a is an isomorphism from D to £)' such that a*7 = (a a )~ for all a £ D, and T G GUn(D', K). Conversely, any map of the above form from VHn(D) to VHn(D') is bijective and both the map and its inverse map preserve the adjacency of pairs of points. □ More generally, let n be an integer > 1, 6 > 0, and

(

0

/W

\

-/(») o

K =

I

(6.85)

A/

where A is a 8 x 6 skew-hermitian matrix, which is definite, i.e., xA*x = 0 for x € D^ implies x = 0. We define the unitary group U2n+8{D,K) and the generalized unitary group GU2n+6{D,K) of degree 2n + 6 over D with respect to if by

U2n+S (D,K) === {T€ {T€ GL^+siD^TKT GL^+siD^TKT 2n+S(D,K)

=== K} K]

and

GU2n2n+s (D,K) === {Te aK, where
6.9. Application to Graph Theory

363

totally isotropic subspaces, and call the subspaces in Nn(K) points. Clearly, when 6 = 0, Nn(K) = VHn(D), the dual polar space of type 2A2n-i- When 8 = 1, Nn(K) is called the dual polar space of type 2A2n. Define two points W1 and W2 in Nn(K) to be adjacent, if dim(Wi + W2) = n + 1. Then we have the following generalization of Theorem 6.49. Theorem 6.54: Let D be a division ring which possesses an involution — and assume that Assumptions 1° and 2° hold. Let n be an integer > 3, 6 > 0, and K be the (2n + S) x (2n + 6) skew-hermitian matrix defined by (6.85). Let A be a bijective map from Nn(K) to itself and assume that both A and A'1 preserve the adjacency. Then A is the form W^W'T

for all W G Nn{K),

(6.86)

where a is an automorphism of D which commutes with — and T is an element of GU2n+s(D,K). Conversely, any map of the form (6.47) from Nn(K) to itself is bijective and both the map and its inverse map preserve the adjacency. For the proof of Theorem 6.49, cf. Dieudonne 1955.

6.9

Application to Graph Theory

Let D again be a division ring which possesses an involution —. Now we call the points /Hn(D) vertices and define two vertices Hi and H2 to be adjacent if rank(£Ti — H2) = 1. Then we obtain the graph of hermitian matrices. Denote this graph by T(Hn(D)). From Proposition 6.5 we deduce Proposition 6.55: Let D be a division ring which possesses an involution such that Assumptions 1° and 2° hold, and n be an integer > 2. Then T(Hn(D)) is a connected graph of diameter n. □ Theorem 6.37 can be interpreted as follows. Theorem 6.56: Let D and D' be division rings which possess involutions — and ~ , respectively, and assume that Assumptions 1° and 2° are satisfied for both D and D'. Let n and n' be integer > 2. When n = 2, we assume that D is a field, and when n' = 2 we assume that D' is a field. If there

364

Chapter 6. Geometry of Hermitian

Matrices

is a graph isomorphism from the graph of hermitian matrices T(Hn(D)) to T(H'n(D')), then n = n', D is isomorphic to D', and the graph isomorphism is of the form (6.77) Z h - ) f t t T Z ' P + Ho, for all Z G Hn{D), where a G F'\ P G GLn{D'), H0 G ? 4 ( £ ' ) such that a* = (a a )~ for all a £ D. Conversely, if a is an isomorphism from D to D' such that a a — (aa)~ for all a G D, then any map of the form (6.38) is a graph isomorphism from T(Hn(D)) to T(Hn(D')). □ Corollary 6.57: Let D be a division ring which possesses an involution — such that Assumptions 1° and 2° are satisfied and n be an integer > 2. When n = 2, assume further that D is a field. Then the group of graph automorphism of the graph T(Hn(D)) consists of the graph automorphisms of the form (6.2). □ Similarly, we call the points of V7in(D) vertices and define two vertices W\ and W2 to be adjacent if W\ + W2 is of dimension n + 1. Then we obtain the graph of the protective space o f n x n hermitian matrices over D, which is denoted by T(VHn(D)) and is also called the dual polar graph of type 2 A2n-i- From Propositions 6.45 and 6.47 we deduce Proposition 6.58: Let D be a division ring which possesses an involution — such that Assumptions 1° and 2° are satisfied, and n an integer > 1. Then the graph T(V7in(D)) is connected, distance-transitive, and with diameter

n.

□

Theorem 6.53 can be interpreted as follows. Theorem 6.59: Let D and D' be division rings which possess involutions — and ~ , respectively, such that Assumptions 1° and 2° are satisfied for both D and £)', and let n and n' be integers > 2. When n = 2, assume that D is a field, and when n' = 2, assume that D' is a field. If there is a graph isomorphism from the graph T(VHn(D)) to T(VHn>{D')), then n = n', D is isomorphic to D\ and the graph isomorphism is of the form (6.84) W .—> WaT

for all W G

PKn{D),

6.10.

Comments

365

where a is an isomorphism from D to D' such that a a = (a*)~ for all a G D and T G GU2n(D,K). Conversely, any map from T(VHn(D)) to T(VHn(D')) of the form (6.84) is a graph isomorphism. □ Corollary 6.60: Let D be a division ring which possesses an involution — such that Assumptions 1° and 2° are satisfied and n be an integer > 2. When n = 2, assume further that D is a field. Then the group of graph automorphisms of T(VHn(D)) consists of graph automorphisms of the following form W ■—> WaT

for all W G

VHn(D),

where cr is an automorphism of D such that aa = ~a? for all a G D and TeGU2n(D,K). D More generally, for the (2ra + 6) x (2n + <5) skew-hermitian matrix if defined by (6.85) call the elements Nn(K) vertices and define two vertices W\ and W2 to be adjacent if dim (Wi + W2) = n +1, then we obtain a graph which is denoted by T(Nn(K)), and Theorem 6.54 can be interpreted in graph theory language as follows. Theorem 6.61: Let D be a division ring which possesses an involution — such that Assumptions 1° and 2° are satisfied, n be an integer > 3, 6 > 0, and if be a (2n + 6) x (2n + 6) skew-hermitian matrix defined by (6.85). Then the group of graph automorphisms of T(Nn(D)) consists of graph automorphisms of the form (6.86) W H—> WaT

for all W G

Nn(K),

where a is an automorphism of D such that aa = aa for all a G D and TeGU2n+6(D,K). □

6.10

Comments

Theorem 6.4, i.e., the fundamental theorem of the geometry of hermitian matrices was proved by the author, cf. Wan 1995a,b. When the ground division ring is the complex field, it was proved by L. K. Hua by assuming

366

Chapter 6. Geometry of Hermitian

Matrices

more conditions, cf. Hua 1945a,b. When the ground division ring is a finite field, it was proved by A. Ivanov and S. Shpectorov, cf. Ivanov and Shpectorov 1991. Sections 6.1 - 6.7 follow Wan 1995b, 1996. Theorem 4.49, i.e., the fundamental theorem of the projective geometry of hermitian matrices, and its generalization, Theorem 6.54, is due to J. Dieudonne, cf. Dieudonne 1955. The results of Sections 6.9 and 6.10 are immediate.

Bibliography [1] Artin, E. (1957) Geometric Algebra, Wiley, New York. [2] Baer, R. (1952) Linear Algebra and Projective Geometry, Academic Press, New York. [3] Bannai, E. and Ito, T. (1984) Algebraic Combinatorics I: Association Schemes, Benjamin/Cummings, London. [4] Brouwer, A. E., Cohen, A. M., and Neumaier, A. (1989) Distanceregular Graphs, Springer. [5] Chow, W.-L. (1949) On the geometry of algebraic homogeneous spaces, Annals of Mathematics 50, 32-67, [6] Deng, S.-T. and Li, Q. (1965) On the affine geometry of algebraic homo­ geneous spaces, Ada Mathematicae Sinica 15, 651-663, (In Chinese). English translation: Chinese Mathematics 7 (1965), 378-391. [7] Dieudonne, J. (1951) Algebraic homogeneous spaces over fields of char­ acteristic two, Proceedings of American Mathematical Society 2, 295304. [8] Dieudonne, J. (1955) La geometric des groupes classiques, Springer. Deuxieme edition, 1963. Troisieme edition, 1971. [9] Egawa, Y. (1985) Association schemes of quadratic forms, Journal of Combinatorial Theory, Series A 38, 1-14. [10] Gao, J., Wan, Z., Feng, R., and Wang, J. (1996) Geometry of symmetric matrices and its applications IH, to appear in Algebra Colloquium 3. 367

368

BIBLIOGRAPHY

[11] Hua, L. K. (1945a) Geometries of matrices I. Generalizations of von Staudt's theorem, Trans. Amer. Math. Soc. 57, 441-481. [12] Hua, L. K. (1945b) Geometries of matrices Ilm Arithmetical construc­ tion, Trans. Amer. Math. Soc. 57, 482-490. [13] Hua, L. K. (1946a) Geometries of symmetric matrices over the real field I, Dokl. Akad. Nauk. SSSR (N. S.) 53, 95-97. [14] Hua, L. K. (1946b) Geometries of symmetric matrices over the real field H, Dokl. Akad. Nauk. SSSR (N. S.) 53, 195-196. [15] Hua, L. K. (1947a) Geometries of matrices H. Study of involutions in the geometry of symmetric matrices, Trans. Amer. Math. Soc. 6 1 , 193-228. [16] Hua, L. K. (1947b) Geometries of matrices IH. Fundamental theorems in the geometries of symmetric matrices, Trans. Amer. Math. Soc. 6 1 , 229-255. [17] Hua, L. K. (1949) Geometry of symmetric matrices over any field with characteristic other than two, Ann. of Math. 50, 8-31. [18] Hua, L. K. (1951) A theorem on matrices over a sfield and its applica­ tions, Ada Math. Sinica 1, 109-163. [19] Hua, L. K. and Rosenfeld, B. A. (1957) Geometry of rectangular matri­ ces and their application to real projective and non-eucliden geometry, Scientia Sinica 6, 995-1011. [20] Hua, L. K. and Wan, Z. (1963) Classical Groups, Shanghai Science and Technology Press, Shanghai, (In Chinese). [21] Huo, Y. and Wan, Z. (1993) Non-symmetric association schemes of symmetric matrices, Ada Mathematicae Applicatae Sinica 9, 236-255. [22] Huo, Y. and Zhu, X. (1987) Association schemes with several associate classes of symmetric matrices, Ada Mathematicae Applicatae Sinica 10, 257-266, (In Chinese).

BIBLIOGRAPHY

369

[23] Ivanov, I. I. and Shpectorov, S. V. (1991) A characterization of the association schemes of Hermitian forms, J. Math. Soc. Japan 43, 2548. [24] Jacobson, N. (1968) Structure and Representations American Mathematical Society.

of Jordan Algebras,

[25] Jacobson, N. (1985) Basic Algebra, Vol. I, Second Edition, Freeman, New York. [26] Liu, M. (1965) A proof of the fundamental theorem of the projective geometry of symmetric matrices, Shuxue Jinzhan 8, 283-292, (In Chi­ nese). [27] Liu, M. (1966) Geometry of alternate matrices, Ada Math. Sinica 16, 104-135, (In Chinese). English translation: Chinese Mathematics 8(1966), 108-143. [28] Pei, D. (1964) A proof of the fundamental theorem of the projective ge­ ometry of rectangular matrices, A Collection of Papers Celebrating the Fifth Anniversary of the Chinese University of Science and Technology, 99-110, (In Chinese). [29] Tits, J. (1974) Buildings and BN-Pairs of Spherical Type, Lecture Notes in Math. No. 386, Springer Verlag, Berlin. [30] Veblen, and Whitehead, (1932) Foundations of Differential

Geometry,

Cambridge Tracts 29. [31] Wan, Z. (1961) A proof of the automorphisms of linear groups over a field of characteristic 2, Ada Mathematicae Sinica 1 1 , 380-387, (In Chinese). English translation: Scientia Sinica 11(1962), 1183-1194. [32] Wan, Z. (1965) Notes on finite geometries and the construction of PBIB designs VI, Some association schemes and PBIB designs based on finite geometries, Scientia Sinica 14, 1872-1876. [33] Wan, Z. (1992a) Finite geometries and Block designs, Sankhya: The Indian Journal of Statistics, Special Volumn 54, 531-543.

BIBLIOGRAPHY

370 [34] Wan, Z. (1992b) Introduction dent litter at ur, Lund.

to Abstract and Linear Algebra, Stu­

[35] Wan, Z. (1993) Geometry of Classical Groups over Finite Fields, Studentlitteratur, Lund. [36] Wan, Z. (1994a) Geometry of symmetric matrices and its applications I, Algebra Colloquium 1, 97-120. [37] Wan, Z. (1994b) Geometry of symmetric matrices and its applications I , Algebra Colloquium 1, 201-224. [38] Wan, Z. (1995a) The graph of binary symmetric matrices of order 3, Northeastern Mathematical Journal 1 1 , 1-2. [39] Wan, Z. (1995b) Geometry of Hamiltonian matrices and its applications I, Algebra Colloquium 2, 167-192. [40] Wan, Z. (1996) Geometry of Hamiltonian matrices and its applications H, to appear in Algebra Colloquium 3. [41] Wan, Z., Dai, Z., Feng, X., and Yang, B. (1966) Studies on Finite Geometry and the Construction of Incomplete Block Designs, Science Press, Beijing, (In Chinese). [42] Wan, Z. and Wang, Y. (1962) Discussions on "A theorem on matri­ ces over a sfield and its applications', Shuxue Jinzhan 5, 325-332, (In Chinese). [43] Wang, Y. (1980) Association schemes with several associate classes of matrices, Ada Mathematicae Applicatae Sinica 3, 97-105, (In Chinese).

Index a system of coordinates, 67 action of a group, 28 addition of matrices, 13 addition of vectors, 1, 9, 10 adjacence, 90, 124, 153, 154, 157, 182, 198, 203, 217, 288, 296, 358, 363, 364, affine equivalence, 51 affine geometry, 51 affine group, 51 affine r-flat, 45 affine transformation, alternate matrix, 39 anti-automorphism of

cogredience transformation, 36, 39 coherence, 90, 124 column rank, 21 column vector, 9, 12 column vector space, 9 connected graph, 153 consistent system of linear equations, 34, 47 coordinate(s), 45, 67

210, 211, 302, 306, 365

diagonal matrix, 16 diameter, 153 Dickson invariant, 200 dimension formula, 7, 50, 69 dimension of a subspace, 6 dimension of a vector space, 10 dimension of an r-flat, 45, 68 distance, 91, 127, 159, 184, 203,

51 a division

ring, 17, 35 anti-isomorphism of division rings, 66^ 80 arithmetic distance, 89, 124, 157, 182, 217, 288, 306, 358 automorphism of a division ring, 55, 78

219, 289, 308, 359 distance-transitive graph, 153 division ring, 1 dual flat, 75, 178, 286, 357 dual polar graph of type 34.2n-i? 364 dual polar graph of type C n , 302 dual polar graph of type Z>n, 212 dual polar space of type 2A2n, 363 dual polar space of type 34 2n -i, 358 dual polar space of type Bn, 198

basis, 2, 3, 10 block form of a matrix, 18 block of a matrix, 18 center, 36, 71, 306 371

INDEX

372 dual dual dual dual

polar space of type C n , 288 polar space of type Dn, 181 polar space of type 2Dn+i, 198 subspace, 33, 178, 286, 356

echelon matrix, 26 echelon normal form, 26 equation of a hyperplane, 47, 68 equivalence of matrices, 23 Erlangen Program, 51, 71 finite dimensional left vector space, 10 finite dimensional right vector space, 11 finite point, 74, 80, 124, 181, 288, 358 fixed field, 36, 306 fundamental theorem of the affine geometry, 54 fundamental theorem of the geometry of alternate matrices, 158 fundamental theorem of the geometry of hermitian matrices, 307 fundamental theorem of the geometry of rectangular matrices, 90 fundamental theorem of the geometry of symmetric matrices, 218 fundamental theorem of the irreducible space, 204

fundamental theorem of the one-dimensional projective geometry, 85 fundamental theorem of the projective geometry, 77 fundamental theorem of the projective geometry of m x (m + n) matrices, 143 fundamental theorem of the projective geometry of alternate matrices, 185 fundamental theorem of the projective geometry of hermitian matrices, 360 fundamental theorem of the projective geometry of symmetric matrices, 290 general linear group, 20 generalized orthogonal group, 185, 198, 211 generalized orthogonal matrix, 185, 211 generalized projective transforma­ tion, 84 generalized symplectic group, 289 generalized symplectic matrix, 289 generalized unitary group, 360, 362 generalized unitary matrix, 359 generator, 3 graph, 153 graph automorphism, 153 graph isomorphism, 153 graph of m x n matrices, 153

INDEX graph of n x n alternate matrices, 211 graph of hermitian matrices, 363 graph of symmetric matrices, 296 graph of the irreducible space, 213 graph of the left Grassmann space, 154 graph of the projective space of n x n alternate matrices, 212 graph of the projective space of n x n hermitian matrices, 364 group of motions, 89, 124, 157, 217, 306 half dual polar graph of type Dn, 213 harmonic set, 81 hermitian matrix, 35 homogeneous coordinate(s), 74, 124, 181, 288, 358 hyperplane, 45, 66 hyperplane at infinity, 74 idempotent matrix, 119, 283 identity matrix, 16 improper generalized orthogonal matrix, 204, 211 improper orthogonal matrix, 199, 200, 210 incidence relation, 45, 66 independent set of points, 69 independent system of linear equations, 34, 47 index, 177, 179 intersection of flats, 50, 68

373 intersection of subspaces, 7 inverse matrix, 19 invertible matrix, 19 involution of a division ring, 35, 305 irreducible space, 201, 210 isomorphism of division rings, 66,80 isomorphism of Jordan rings, 122, 281 isomorphism of vector spaces, 11 join of Jordan Jordan Jordan

flats, 50, 68 isomorphism, 122 multiplication, 122 ring, 122, 281, 355

left affine space, 45 left Grassmann space, 123 left inverse, 20 left invertible matrix, 20 left projective line, 80 left projective space, 67 left vector space, 1, 10 linear combination, 2, 3 linear dependence, 2 linear independence, 2 line, 45, 66, 95, 130, 165, 230, 251, 320, 322, 323, 345, 347, 348 matrix, 12 matrix representation of a subspace, 28 maximal set, 159

INDEX

374 maximal set of rank 1, 93, 127, 222, 308 maximal set of rank 2, 97, 224, 244, 270, 311 maximal totally isotropic subspace, 179, 287, 357, 362 maximal totally singular subspace,

plane, 45, 66, 104, 167

180, 198 method of intersection and join, 60 modular law, 8 multiplication formula of matrices in block form, 19 multiplication of matrices, 15 multiplicator, 185

principal diagonal, 16

negative of a matrix, 13 non-homogeneous coordinate(s), 74, 80, 124, 144, 181, 187, 288, 291, 358, 361 nonalternate symmetric matrix, 41 nonzero subspace, 3 norm map, 305

projective line, 80

orbit, 29 orthogonal group, 178, 180, 198 orthogonal idempotent matrices, 119, 283 orthogonal matrix, 177, 179 orthogonal polarity, 178

point, 45, 66, 89, 123, 124, 157, 181,198, 217, 306, 358, 363 point at infinity, 74, 80, 124, 181, 288, 358 primitive idempotent matrix, 119, 283 principle of duality, 75 product of matrices, 15 projective r-flat, 66 project ive equivalence, 71 projective general linear group, 71 projective geometry, 71 projective space of (m + n ) x n matrices, 138 projective space of m x (m + n) matrices, 124 projective space oinxn

alternate

matrices, 181 projective space oinxn

hermitian

matrices, 358 projective space oinxn

symmetric

matrices, 287 projective transformation, 70, 71 proper generalized orthogonal group, 204, 211

parallel flats, 49 parametric equation, 95, 104 parametric representation, 49, 70 perfect field, 300 permutation matrix, 23

proper generalized orthogonal matrix, 204, 211 proper orthogonal group, 200, 210 proper orthogonal matrix, 199, 200, 210

INDEX

375

r-flat, 45, 66 rank, 24 reduced maximal set of rank 2,318 replacement theorem, 4

solution space, 31 solution vector, 30 space of m x n matrices, 89 space of n x n alternate matrices,

right affine space, 54 right Grassmann space, 138 right inverse, 20 right invertible matrix, 20 right projective space, 75 right vector space, 9 row equivalence of matrices, 26 row rank, 21 row vector, 1, 12 row vector space, 1

157 space of n x n hermitian matrices, 306 space of n x n symmetric matrices, 217 space of rectangular matrices, 89 submatrix, 17 subspace, 3 sum of matrices, 13 sum of subspaces, 7 sum of vectors, 10 symmetric matrix, 37 symmetrized multiplication, 122,

scalar, 10, 14 scalar multiplication of a matrix by a scalar, 14, 15 scalar multiplication of a vector by a scalar, 1, 9, 10 scalar product of a matrix by a scalar, 14 scalar product of a vector by a scalar, 10 self-dual flat, 179, 357 self-dual subspace, 179, 287, 357 semi-automorphism of a division ring, 82 semi-isomorphism of division rings, 82 semi-isomorphism of rings, 120 skew-symmetric matrix, 39 skew lines, 49 solution, 30

281 symplectic group, 286 symplectic matrix, 285 symplectic polarity, 286 system of linear equations, 29 system of linear homogeneous equations, 30 system of linear non-homogeneous equations, 30 total matrix ring, 118 totally isotropic subspace, 179, 198, 287, 357, 362 totally singular subspace, 180, 198 trace map, 36, 305 transitive set, 29 transpose of a matrix, 17

376 unitary group, 356, 362 unitary matrix, 356 unitary polarity, 357 vector, 10 vector space, 1 vertex, 153, 154, 211, 296, 302, 363, 364, 365 zero matrix, 13 zero subspace, 3 zero vector, 2, 10

INDEX