Methods of Matrix Algebra

Au,

w,~Au,= 1

w,+Au,

=

1

... w ~ - ~ A u=, 1 (55)

90

111.

w,tA w;-,A

= Aw,f

w,tA

+

= Awi-,

... = Awl+

+ w,?

A ~ w ,= A*w, A~w,-, = A*w,-,

+ + w2

W,

...

A ~ w ,= A*w,

w l ,..., w,,

W, t

(57)

p

ul , ..., ul, .

k

A At,

( p - k). Eq.

AX, = 0, AX, = x l ,

AX, = 0 AX, = x3

At,

in

ui

p, p,

wi

18.

w1 , w 2 ,wg ,

91

At,

w4

18. VARIATIONAL DESCRIPTION O F T H E EIGENVECTORS A N D EIGENVALUES

As on

x P(X) =

by

(xtAx)/(xtx)

(58)

(x~x)

(xtAx)

(x~x) x

=

p(x)

0.

Raleigh quotient.

p(x)

x ?

x 6x.

p(x

+ 6x) = p(x) no

x.

do

p(x) At

=

A A

92

111.

a

on A. P(x

=

+

+

+

(x Sx)+A(x Sx) (x Sx)t(x 6x) = P(X)

+

+

(59)

(59) (58), 6x

Sxt,

+

(x~x){(SX)~AXxtA(Sx)} = (xtAx){(Sx)tx

+ xt(6x)}

+ { x ~ A- P(X)X~} SX = 0

(Sx)t{Ax - P(X)X}

(60)

x

do

+

6x,

Sx.

(60) (Sx)t(Ax - P(X)X}

-

( x ~ A- P(X)X~}SX

=

0

(61)

(60) (Sx)t{Ax - P(X)X} 0 { x ~ A- P(x)x~}SX = 0 =z

AX - P(X)X

=

0

x ~ A -P(x)x~= 0

(63)

by A ~x P*(X)X

=0

(64)

(64)

(62) A

x

p(x),

At

p(x)

A

At

(64)

18.

93

VARIATIONAL DESCRIPTION

A

A

At. so

of all

A

xtAx x.

by

P(X) = (x~KAx)/(x~Kx)

K

(Kt

=

(65)

K),

(64),

K.

(K-’AtK)x

- P*(X)X = 0

(66)

A

(K-’AtK).

A, A

by

I x

(y, x)

x # 0,

I

y E.

A y,

p(x, y) x,

94

111. EIGENVALUES A N D EIGENVECTORS

x

p(x) A, A

=

A,

+ €A1,

E

x,

x2

A,, A

y, yz by x1 x2 . y,

y,

good

Exercises

1. A =

(;

a b O a b) c a

2. 0

0

3.

4.

A=(

0

95

EXERCISES

0

x.

no

A

? ?

you

5.

A. At. 6.

7.

A=

(

*

e

yy!:22)

%-1 “n

***

0 0

0 0 ai

a1 =

0, an # 0 ?

96

111. EIGENVALUES A N D EIGENVECTORS

8. If A

AX

n x n

=

XA

X

n x n

A

of of

A

9.

B

n x n

of (A

+ B)

A.

A, .

10. 0

0 1

1 0

...

0

0 -aT,-l

0 -anp2

0 0 01

... ...

-an-4

...

... ...

0 -anp3

7

1 --a,

n

n x n rational canonical form. on 11.

s

II

+

I

I

+

1{1

+

=

+ ...

s

I A

+

I

=

1

+ *..)

n x n =0

X. 12. If p(A)

B

A AB

A, =

0

n x n

A,

CHAPTER I V

Hermitian, Unitary, and Normal Matrices

by

by

no by

go

by

good

1. ADJOINT RELATION

97

98

IV. HERMITIAN, UNITARY, AND NORMAL MATRICES

adjoint,

on,

(x, y), functional,

A

y,

Ay, by

A#,

x

on

by of

by Definition. Given a linear vector space with an inner product relation dejined. If A is a matrix operator in this space, then A# is its adjoint i f (x,

for all x and y in the space.

Ay)

=

(A% y)

(1)

2.

A,

99

RULE OF COMBINATION

A# y,

x (A#x)ty = xtA#ty

A#

y

=

=

xtAy

(3)

(4)

At

Ay

(A#ty),

x

K: (5)

< x , Y >= xtKy

K y

x (A#x)tKy = xtA#tKy

A#

= xtKAy

(6)

(7)

= K-lAtK

by 2. RULE O F COMBINATION

by <x,ABY)

x =

<(AB)#x,Y)

=


=

(B#A#x, y)

(AB)# = B#A#

(ABy): (8)

(9)

100 At

(AB)t

=

BtAt

(10)

to

3. T H E BASIC TYPES

of Eq.

A,

by Eq. (4),

A Definition. If A is equal to its adjoint under the unitary inner product, then A is hermitian: A = At (11)

Definition. If A is the negative of its adjoint under the unitary inner product relation, A is skew-hermitian: At

=

-A

(12)

Definition. If the adjoint to A under the unitary inner product is equal to the reciprocal of A, then A is unitary: At =A-'

or

AAt = I = A t A

of

Definition. If A commutes with its adjoint under the unitary inner product, A is normal: AAt = AtA (14)

A

3.

101

BASIC TYPES

by (x, Y>

=y 'x

(15)

(T)

symmetric (atj = a j i ) .

A A

skew-symmetric.

AT orthogonal.

AT,

normal.

(xy): A, A#:

XtY

XTY

At

AT

of

A# = A# = AA# = A#=

A A-1 A#A -A

n x n

A

B (AB)t of

= BtAt = B-lA-1 = (AB)-'

C , ...} of

property),

(16)

group group

102 n x n n x n of

As

4. DECOMPOSITION INTO HERMITIAN COMPONENTS

A A, :

A,

+ At)

A,

=

&(A

A,

=

-j&(A - At)

A

A,

= A,

(17)

+ jA,

A,

A,

of A

jA, of

A

1 x

5. POLAR DECOMPOSITION

of

A,

U,

H,

H,

U, , A

= HIUl = U,H,

(21)

We At

= U tH t = U1

1

1

'Hl

6.

STRUCTURE

H,

103

NORMAL MATRICES

H, H,2

= HlU1U;lHl = AAt

(23)

H,2

= H2U;1U2H, = ATA

(24)

A, (AAt)t = AAt (AtA)t = AtA

H,

H,

H,

H, .

A

H, U,

= H;'A

(26)

U,

= AH;'

(27)

A H,

U, U, ,

(27)

Eq.

F Eq. H, # H,

6. STRUCTURE O F NORMAL MATRICES

So

a

U1#

104

IV. HERMITIAN, UNITARY, AND NORMAL MATRICES

by

Definition. Given an inner product relation and a subspace U , the orthogonal complement of U is the maximal subspace V , such that if x is anv nactor in V , then for all vectors y in U

(x, Y>

=0

(29)

V U.

(au

+ bv)

u

V

V,

v

V.

Theorem 1 . If the inner product relation is proper, a subspace U and its orthogonal complement V are disjoint-i.e. have no vector in common, except the null vector.

U

by

XI

, x2 , ..., xk .

U.

U

by (y, y)

V.

(y, xi) y

U

V

6.

105

STRUCTURE OF NORMAL MATRICES

Theorem 2. If U is a proper subspace and V its orthogonal complement under a proper inner product, then V is nonvoid.

U U. U, U,

projection on U ,

U,

V.

V A

U

A (Ax).

U

x

invariant

Theorem 3. If U is invariant for A, then V ,its orthogonal complement under a given inner product, is invariant f o r A#, the adjoint to A under the given inner product.

x

A, Ax

U

U, by

V.

y


A#y

U

(32)

U,

V.

A#.

V

Theorem 4. If, under a given proper innerproduct relation, A commutes with its adjoint A#, then A is semisimple and we can so choose its eigenvectors that they are orthogonal by pairs under the given inner product relation.

by

by by

n

A

1,

111, A

x1 : AX,

=h,~,

(33)

106

I V . H E R M I T I A N , U N I T A R Y , A N D N O R M A L MATRICES

A# cyclic subspace of A#. Y l = XI

yz = A#x,

...

yk

=

A#yk-,

= (A#)k-lX1

(34)

y1 , ..., y k , on y1 , ..., y,( ,

yk+,

yi .

yi . y, , ...,y, , 1,

S, A#

by Ahl

A

(36)

= plu,

S, A A

111,

u1 :

A#,

,

ul,

A,.

A#: A(A#)lJxl= AA#(A#)J’-’xl= A#A(A#)”-’x, -

x1

A,,

A

A#, ( A # ) ~ X ,

U, U,

A

A

p S, ul

(37)

... = (A#)”Axl = Al(A#)”xl

by

2,

A by u,, V, A A#.

A#.

A. by

3,

107

7. A# A, V ,

V,

x,, V,

A.

2,

by

V,, A#,

A#,

ul . u2 ul .

A

A#

U,, V, .

by

u2 ,

by

U, ,

ul, ...,u,

A

so

u2 .

x,

A,

u,

u1

x2 . A#.

A,

by of

A#.

V, V,

A, A#, A

A#,

u,+,

V/;,

u1 ..., uk uk

n

n

A

A#.

A,

A#, 7. T H E CONVERSE T H E O R E M

is

Theorem 5 . If the matrix A has a complete set of eigenvectors u, ,u2,...,u, which are orthogonal by pairs:

(ui,uj)

= 0,

i#j

(38)

then A is normal and AA#

by

= A#A

(39)

ui

A#.

y . = A#u. - A,* o t ui

(40)

108 do

yi

ui

of yi (uj , y i )

= (ti,

=

,A h i )

- I\i*(uj,u?)

(Auj ,ui) - Ai*(uj, ui)

= h,*(u, ,Ul) - h,*(u,, Ui) =

-

i=j

X,*)(U,,

Ui) = 0

(Xi*

by

(41) -

Xi*).

of

ui

Eq.

uj

y, on i,

yi

Ahi

ui

(42)

= Ai*ui

Xi*.

A#

x.

ui

x

, (43)

=zaiui 1

A#A~ =~

#

~

z

a

~

i

= A#

z

aiXiui

i

=

zai~i~i*ui 1

x

A exterior exterior

interior

on

u

~

8.

109

HERMITIAN MATRICES

interior

8. HERMITIAN MATRICES

by Theorem 6 .

If A is self-adjoint, its eigenvalues are real. x1 , x2,..., x,

:

AX^ (xi, Axj)

= Xj(xi, xj) == = &*(Xi

,

( A # x ~xj) ,

=

AX^, xj) (47)

Xj) = 0

(hi* - Xj)(Xi,

i

(46)

= hixi

Xj) =

0

=

hi*

hi

(49)

= hi

(48) 4 A A

110

IV. HERMITIAN, UNITARY, A N D NORMAL MATRICES

9. U N I T A R Y MATRICES

Theorem 7. if A is a matrix such that its adjoint equals its reciprocal, then its eigenvalues have unit modulus.

xl,x2,...,x,

AX^ , Axj)

==

Xi*Xj(xi, xj)

=

( x i , A#Axj)

-= (Xi

>

=

(xi, Ix,)

xi>

- 1 ) ( X i , Xi) = 0

i

==

(xixi)# 0 by

j,

hi i2

=

1,

10. GENERAL (PROPER) I N N E R P R O D U C T

of on on

<x,Y)

of

(52)

= xtKy

K

(x, x) x y

so

x K-orthogonal, (x, y)

=x

~ yK = o

x <x,AY) =

(53)

y (54)

10.

GENERAL (PROPER) INNER PRODUCT

111

xtKAy

(55)

x

=

(A#x)tKy = x+A#tKy

y KA

K-adjoint

(56)

= A#tK

A A#

= K-'AtK

(57)

K

Definition.

A matrix A is K - n o r m a l AK-lAtK

Definition.

=

if it commutes with its K-adjoint:

A matrix A is K-hermitian A KA

(58)

K-lAtKA

if it equals its K-adjoint:

= K-lAtK

(59)

= AtK

Definition. A matrix A is K-skew-hermitian if A equals the negative of its-K-adjoint:

Definition.

A matrix A is K - u n i t a r y if its K-adjoint equals its

reciprocal: K-lAtK AtKA

(61)

= A-1

-

=K

(62)

Theorem 8. A K-normal matrix A iias a complete set of eigenvectors that can be chosen to be K-orthogonal in pairs, and conversely. Theorem 9. The eigenvalues of a K-hermitian matrix are real, and those of a K-unitary matrix are of unit magnitude.

on so.

112

IV. HERMITIAN, UNITARY, A N D NORMAL MATRICES

As to

K

of

Exercises

1.

a

a

b

2.

2

0

2,

3. (A,

+ jA,)

n x n

A,

A,

A

b,

113

A,

4.

AAt

AtA

A ?

A

A

K

5.

6.

A

7.

A

n x n

II Ax I1 x,

/3

01

(aA + @At)

II x II

=

=

I1 Atx I1

(x, X

Y 2

A ?

8.

A A

(A2 = A), (A2 = A

A

9.

10.

A

At

by

11. A B (AB + BA)

(AB - BA) A

B ?

A

12.

x 13.

A2x = 0, A

Ax

=

0.

B

B A-ABI = O

A

114

I V . HERMITIAN, UNITARY, A N D NORMAL MATRICES

by

p = - xtAx xtBx

p

T2= I

14. A

15. A

?

involution.

At

16.

A

A U

=

U

+ I)(A

-

I)-'

(U - I) A

A (Hint: (A + I) (U I). ?)

+

(A

=

(U - I)-'(U

Cuyley transforms

U

w = (z

(A

-

+ I)

I)-l

+

-

do (U - I)-l

CHAPTER V

Change o f Basis, Diagonalization, and the Jordan Canonical Form Jordan canonical form.

by

by

do

no

by similarity transformation.

change of basis.

1. CHANGE O F BASIS A N D SIMILARITY TRANSFORMATIONS

representation of the abstract vector. As

by

so 115

116

CHANGE OF BASIS, DIAGONALIZATION

by

A on by

? on

x

u.

S. x = su.

u

(1)

x

on on S

on S.

S,

S

u

= s-'x

(2)

y

= AX.

(3)

as

2.

117

EQUIVALENCE TRANSFORMATIONS

A on

by

(1). ? a

x

=

su,

(4)

y = sv

(3),

by S-l,

v = (S-'AS)u

A =

=AU

(5)

A'

S-lAS

(6)

A

(3) S similarity transformation.

by S

similar.

on good

A, A'

on

by no

A

S

A'

by a collineatory

transformation. 2. EQUIVALENCE TRANSFORMATIONS

x (3),

y,

118

CHANGE OF BASIS, DIAGONALIZATION

x

y

A

do do

by x

=

u

su,

=

s-lx

(7)

v = T-’y

y = Tv,

T

S

A equivalence transformation. equivaZent.l

3. CONGRUENT A N D CONJUNCTIVE TRANSFORMATIONS

of

A

B

by B

(’)

=

STAS

A

B

=

STAS

(9)

congruent.

by

B

(10)

conjunctive.

There is some variation in the literature on names for the various types of transformations and on their definitions. We are here following the usage of G. A. Korn M. Korn, “Mathematical Handbook for Scientists and Engineers,” McGraw-Hill, and New York, 1961.

4.

x

119

EXAMPLE

y,

K

by

(x, Y>

x

(1 1)

= xtKy

y,

x on y,

y. (6),

by

(6) (x, y) = (Su)tK(Sv)= uWKSV

(u,v)

x

(12)

(x, y)

y. (u, V)

K

K,

=~

=

K ’ v

(13)

StKS

by by on by

A

by

(3) (1 4. EXAMPLE

As

120

V. CHANGE OF BASIS, DIAGONALIZATION

111, Eq. (36), on

2

111,

-f)

M'

=

S-'MS

= (cos a,

=

a,

cos a, O

;

a,)

(18)

diag(ej7, e-je)

on

5,

11,

by

)

K = - 1( 0 1 2 1 0

K,

Eq. (15) MtKM

K',

=K

(14):

K

= StKS = Z

1 (0 -- 1

(20)

6.

121

INVARIANCE OF THE EIGENVALUES

parity

by on

on u wave basis. on by

5. GAUGE INVARIANCE

by by

by by

no

by

invariant

principle of gauge invariance.

6. INVARIANCE O F T H E EIGENVALUES UNDER A CHANGE O F BASIS

A by A:

x

AX = AX

122

CHANGE

BASIS, DIAGONALIZATION

(4),

A

(4)

by S-l,

u

=

A u =xu A'

Sx

(23) A.

on h

(4)

by

S

by

k

h on

x,

X

x,

2 AX, = AX,

A,

x2

:

+ x1

(24)

(4), XI

=

su, ,

A'u,

x2 =

= h,

+

su,

Ui

A, u,

x1 A'

u,

2

u, .

A' A A'.

7. INVARIANCE O F T H E TRACE

111,

tr A =

12,

Aii

8.

123

VARIATION OF THE EIGENVALUES

S, Bij =

B

z

(27)

(S-'),p4khShj

kh

S-l

z(s-l)iisik = &,

(28)

i

B

of

k 2

3.

8. VARIATION O F T H E EIGENVALUES UNDER A C O NJ UNCTlVE TRANSFORMATION

A of

K (x,x)

x.

K',

K K'

Kt

=

StKS

=

(StKS)t

z=

StKS

K'

=

=K

StKt(St)t

K,

124

CHANGE

BASIS, DIAGONALIZATION

by

K

S

Sylvester's law of inertia.

szgnature

K' on

K

parity matrix

wave basis. 9. DIAGONALIZATION

A. A

..., A,

A,,

x, , ..., x,

:

A

Aj

xi do by

s = (XI

x2

***

x,)

(33)

A,

S S

S by xk .

x1 , ..., x, S

. ?

S

A on

9. 111,

125

15,

wi

xi. witxi

wi

(34)

= Sij

. S. S

S-’

(33))

by

S-1

A by S, AS = A(xl

X,

x,)

by S-l,

= (A1xl

A,x,

**.

A,x,)

(37)

126

V. CHANGE OF BASIS, DIAGONALIZATION

of A, on

reduced to diagonal form. A.

A

A (33)

S-'AS by

S

(33) by S-l

(35),

S A, A,

S x1

=

A,,

(33)

x, w1

x2.

x,

w2, (38),

A

10. DIAGONALIZATION O F NORMAL MATRICES

xi 2

- 8.. 13

S (39)

(39)

3 -

(33)

xi

(34), S-l,

by

(39,

11.

127

THE HERMITIAN MATRIX

by

s:

s-1

=

S A

S

S A

by

=

S

, A,, ...)S-l

=

S

, A,, ...)St

(40)

by

K by

S

S by

11. CONJUNCTIVE TRANSFORMATION O F A HERM I T l A N MATRlX

K by K

K'by S hi

K'. K":

on

K

K'

K" = DtKD sgnAi

A,

=

=

--1 -

K"

K'

=

1

,

A,, ...)

Xi > 0 Ai
128

V. CHANGE OF BASIS, DIAGONALIZATION

K,

Krr

K”

(41 1

= DtStKSD = (SD)tK(SD)

by no

(41) on

K up on

... . 12. EXAMPLE

5,

11, on

O P \K = (P 0)

P = diag(..., l/f-,

Z = diag( ..., 2-, 2,

,

, ...)

2, , 2, ...)

(43)

(44)

as

fi.

z -z s=(I

I)

(45)

12.

P

129

EXAMPLE

Z

K' By

K" on

+ nfp

= f,,

up

n I

K"'

= (0

0 -1

) on

H " = (I 0) = I

.o

I

on

K"' s1 = utu

-

130

V. CHANGE OF BASIS, DIAGONALIZATION

Zi ui

vi K"

K

H"

13. POSITIVE DEFINITE HERMITIAN FORMS

11,

x

by S(X) =

xtKx

(47)

on s(x)

K

x

s(x) = xtS+-1K'S-'x = (S-'x)tK'(S-'x)

K'

S

y

=

(48)

S-'x y.

yi s ( Y ~ )= Xi

K

hi

Xi

K

(49)

K'.

14. yi yi

131

LAGRANGE’S METHOD

s

.

K

hi y

yi

h

hi

by

.

s

y

y, 14. LAGRANGE’S METHOD

K S(X) = xtKx

= 2xi*xikii ij

K kij

= k,*, ,

kii real

kll # 0.

s1

sl(x) xl. xl, x1 :

=

1

kll

2klikjlxixj* ij

by

132

CHANGE OF BASIS, DIAGONALIZATION

x1 x1

k,,

=

s(x)

xl*

sl(x)

xl*.

x1

xl*.

k,

0,

by

sl(x) si

.IC,

x,*.

no s(x)

kii

on

z

1 ,- -

klixi

(54)

i

y 2 ,y 3 ,

y1 ,

x1

yi .

K

by

K

on

on

kii

k,,

k,, # 0.

=

A,*, # 0.

sI2(x) x1

x2.

=

k,, = K,,

= 0.

2 kilxlxi* +

z

kjg2xi*

s12by

=

0

15.

133

CANONICAL FORM OF A NONSEMISIMPLE MATRIX

x1

s12

x2.

xl

x2 .

x1

x2

and

(57) x2

x1 ,

x1

x2.

s(x)

15. CANONICAL FORM O F A NONSEMISIMPLE MATRIX

We

As by = X1xl = h1x, =

so

x3

x1

+ x1 + X,

(59)

x2

3,

by

on.

wl,..., w, = 6ij

s=

9

x,

9 ...9

(60)

134

V. CHANGE OF BASIS, DIAGONALIZATION

by S

Jki(h,) hi

k, x k,

k, 1

etc.

16.

135

EXAMPLE

S of

ki of on upper Jordan block of order ki .

Jki(Xi) on

lower Jordan block.

S

Jordan canonical form

A.

16. EXAMPLE

As 0

1

A . ( a0

2j

IA

Ax

- A1

1

-2j

2x.

-f)

(67)

= (A2 - 4)2 = 0

us b - 2jc -2jd 2ja d 2jb

+

(68)

2. x=

2. =

0

= 2a = 2b = 2c = 2d

b, c, d )

136

V. CHANGE

BASIS, DIAGONALIZATION

j.

d

=jb

2a

+ b = -2jc b

by 2, =

2x2

+

x2 =

look b - 2jc -2jd 2ja d 2jb

+

= 2a = 2b = 2c = 2d

=

a = 1.

b,

d),

+1 +j d

2a

+ b = -2jc + 1 b = l c =ja

c =ja,

x2

0.

x1 . x1

=j b

17.

POWERS A N D POLYNOMIALS

/1

/2 1 7

=

A MATRIX

137

0

0 -20

y)

0 -2

x1 x2

xl, S,

x3 x2 x4

x4.

x3. A'.

so

A 17. POWERS A N D POLYNOMIALS O F A M A T R I X

so

by Theorem 1 . If we are given two matrices A and B, and change the basis of both by a given similarity transformation to A' and B', then (A'B') is the transformation of (AB).

138

V. CHANGE

BASIS, DIAGONALIZATION

AB, on by isomorphism.2

A B'

= =

SAS-' SBS-'

= (SAS-l)(SBS-')

AB'

SA(S-'S)BS-l = S(AB)S-'

=

AB. A,

D by S : D A =

S-'AS = SDS-' =

SDSp'SDS-1

==

SD2S-I

(73)

by Ak

A

=

SDkS-1

(74)

, A,, ..., An),

D

..., An2)

A,,

D2 = S

a, b,

S

S'

= (a,b),

S'

a', b',

S S

S' is homomorphic S'

(a',b').

A,

A',

n

is

B,C ,

C'

A'B'

=

(AB)'

a'

b

-

(a' b'),

b', (a,b ) isomoqVzic.

17.

139

POWERS A N D POLYNOMIALS OF A MATRIX

Dk =

Azk,

...,Ank)

(75)

D A,O, ..., Ano) 1, ..., 1) = I

Do = =

A0 = SD'JS-1 = SIs-1 = SS-'

(76) =1

m:

a

+ a,x + + a,x"

p ( x ) = a,

**.

A

+ a,A + a2A2+ + a,A"&

p(A) = a,I

* a *

S,

p(D)

+ a1S-'AS + azS-'A"S + + amS-'ArnS + + amDm

S-'p(A)S = u,S-'S = aoI a,D aZD2 =

+

+

***

**.

A (a0

PP)

+

a1h1

+ +

=

0

=

P(Xz),

arnAlrn a0

P(Ai),

.**,

+

a1Az

P(Am)))S-'

-**i

A

n! =

+ ... + amAZm

P(Am))

P(U,

P(A) =

0

k ! ( n - k)!

3

140

V. CHANGE OF BASIS, DIAGONALIZATION

by ij = XSij

Jij

ij

+

si+l.j

Jm

(Jm)ij

=

(J. .)

-0 =

m

if i < j < n

p-i+i

(84)

- 1

1.

+ 1). i Jm

p xp

>

-

1)

up p(x)

p(x)

= u,xm

+

+ +

am$m-l

*.*

a0

$'(x) = dp/dx

p " ( ~= ) d2p/dX2 J(A)

(k -

k x k,

m

( p - 1). (86)

18.

141

THE CAYLEY-HAMILTON THEOREM

18. T H E CAYLEY-HAMILTON THEOREM

Theorem 2. A n y matrix satisjies its own characteristic equation. That is, we consider the characteristic equation of A:

1A

hI I

-

= p(X)

0

(88)

1

where p(A) is the characteristic polynomial in A of A. Then it follows that

P(A) = 0

(89)

A

by

Eq.

p(h,)

= p(h,) =

*-.

p(A) A

(90)

= p(h,) = 0

A,

k

k.

-

P(4

by A,)k

=

p(h) - h,)*g(h)

(91)

k by

p’(A,), ...,pk-l( =

k x k. p { Jk(Ai)}

of

up

(k -

(k -

p(hi),

p(A),

Eq. (91) Eq. (87)

-

A

A Eq.

Eq. (89)

142 n x n

A An

by

A,

n-1 n x n

Ao = I.

n2

19. T H E M I N I M U M POLYNOMIAL

on by do.

of

A, p(x),

p(.) a,, a,,

= a,

+ a,. + + amx"

(92)

.*-

..., a,, p(A)

= u,I

p ( x ) annihilates

+ a,A + + umAm= 0 A.

x, a , ,

minimum polynomial of A.

A A.

~(x)

+(x)

+(x) ~(x),

x

e(x)

r(x)

of ~ ( x ) . A:

+(x) #(A) = B(A)y(A)

v(A)

r(A). r(x)

*(x).

+ r(A)

=0

r(A)

(95)

20.

143

EXAMPLES

A A,

i=l

ki A

A by

by

on distinct

no X

on

A.

distinct

A

p(A)

A,,

m,

up

(m, ( A - hl)ml.

?(A)

by

hi,

Xi.

mi,max

A

A by 20. EXAMPLES

As

111,

(33). h4 = 0, by

4. A2 = 0. ?(A)

=

X2.

2,

no

2 2.

111,

144

CHANGE

BASIS, DIAGONALIZATION

no

A

2. by A4 - 8A2

+ 161 = 0

by 21. SUMMARY

by

on on by by

up

S

145

EXERCISES

Exercises

1. 0

0 - j 0

8 8

A=(;

0 0

0 - j

?

2.

?

5,

3. 4.

I11

n x n

E 8 ;)

(. ..-

-d

--b

-c

-a

rational canonical form. on

on

by

n x n

n,

n,

n x n

5.

A-l A

A, 6.

X

x = (a . j b) IC

d

146

V. CHANGE OF BASIS, DIAGONALIZATION

a, b, c,

X, ( a d + bc),

d

Xn,n sin nO xn=x-sin 0

I

0 =

w = i(u

1 2(a

sin(n - 1)O sin o

+4

+ d), X,

U,(w)

= XU,(w) - Iu,-l(w)

Chebbyscheg polynomial n.

w

U,(w)

=

sin

w)jsin cos-l

w,

7.

8.

A T-lAtT

9.

B, At T-lBtT. AB

A Bt, T-lAT BA A B A

B by T-lBT (T by by

B

C H A P T E R VI

Functions o f a Matrix

f(x),

?

1. DIFFERENTIAL E a U A T l O N S

dx

_ -- -jRx dz

x

z.

no

z.)

t 147

148

VI. FUNCTIONS OF A. MATRIX

R

x

R

n x n

R

z,

(-j) by

S.

no As w:

dE _ - -juLI dz

dI _ dz

L

-

-juCE

C

L

C

z.

by no good by

E x=(;)

R

=w

2 x 2

m

Z

=

dxjdz

I:

2.

149

REDUCTION OF DEGREE

R

(I),

z

x(0)

=

0.

-jRz)

(4).

do

2. R E D U C T I O N OF DEGREE

(1)

R

z,

R

z,

d2u du dz2 - - A d z B u

v

A du -= v dz

-dv_- A v + B u dz

(9)

150

VI. FUNCTIONS OF A MATRIX

by

(1). 3. SERIES E X P A N S I O N

x: m

f(x) =

2

(10)

u,xn

n=O

of

A

Am

A m

A m.

A

on

J:

N,

(N

fN

N

f d A ) = C;4SJS-l)n n=O

=

S

(2

anJn)S-l

n=O

+

3.

151

SERIES EXPANSION

=

, A,, ...)

=

AZn, ...)

fN(A)= S

(z

unAln,

2unAZn,...)I S-’ ..*)>s-l

=

(14)

up

fN(A)

Xi A

***I

=

mi x mi

J,,,i(hi)

Jn

JtJAJ, ...>

=

J,,, ,

JZl(X,)

(15)

by

-+ 00,

k hi. f(x)

off(x)

152

OF A MATRIX

Eq. (1 Eq. Eq. x,, x,

f(x)

f(x) admissable for

by by f(A) =

S

Eq. .-.)>S-l

(16)

by

Eq. f(A)

JnijAi)}

f{Jm,(hl)), f{Jm,(hz)),

=S

**

.)>S-l

mi x mi

\ etc.

/

V.

(4)

4. TRANSMISSION LINE

Eq.

(17)

5.

153

SQUARE ROOT F U N C T I O N

R 0

R2 = (o

e-iRz

Be) = pZI

1

--

= I - jRz

--

(I

+

1

-

p z 2

+

+ j 31!B22R +

2!

(1 - 2!

= I cos Bz

=

2!

1

(19)

1 R2z2 j - R3z3 3!

= I - jRz .

=1

Eq. (3).

1 p4z4 - .:.) +-

4!

1

j-R

Pz

B

-jZ

pz

2

pz

Bz

R.

E(z) = -. E(0)

/3z

E(O) Z(z) = - j Z on

-

jZZ(0)

/3z

flz

+ Z(0) cos fk z

z =0

(21)

of z = 0.

z

e = -p

5. SQUARE ROOT FUNCTION

As

us

z

(22)

154

OF

5. H12= AAt

(23)

H,2

(24)

= AtA

H, ,

H,

of

H,,.

H12

H12

H,,

A x = 0.

of

H12

H,,

x, xtHlzx = x ~ A A ~=x (Atx)t(Atx) = yty

x

y

...,

A?,

A,, A,,

(25)

of H12

...

H12

H12= S{diag(X12,,A,,

...)} S-l

(26)

S{diag(A, , A,, ...)} S-l

(27)

H, H,

=

A, , A,,

S

...

H, go A,, A,,

of

... .

H,

H, H,

U2

A by

A

H,

H, U,

by

(26)

7.

155

EIGENVECTORS

no

A

IV,

H,

H, XII.

6. UNITARY MATRICES AS EXPONENTIALS

As

U

(28)

=

H a

IV)

U

hi

U

=

H

=

ejA2,

...)} S-l

(29)

H ,A 2 , ...)} S-l

(30)

H 10,

S-l

Ht

S

=

,A , , ...)}

=

, A , , ...)} S-l

=

=H

H

hi H

by

A

7. EIGENVECTORS

f(A), A. (4)

156

VI. FUNCTIONS OF A MATRIX

7)

A,

f(hi)

hi

8) f(A).

Theorem 1 . I f f ( x ) is an admissable function of a matrix A, and xiis an eigenvector of A with eigenvalue Xi , then xi is also an eigenvector o f f ( A ) , but with eigenvalue f (A,)..

A A

A f’(A)

A,

1 Theorem 2. If A and B are semisimple, and any eigenvector of A, regardless of how any degeneracies in A are resolved, is also an eigenvector of B, then B can be expressed as an admissable function of A.

A x1

B, A

x2

axl

B

+ /lx2

A,

B A

xi

A

B.

B

hi

/3.

a

Xi.

hi . pi.

pi,

h

pi

hi,

i = i.

A.

=

Xi.

8.

SPECTRUM xk

157

A MATRIX

k # i,

A.

i = k,

B xk

p(A)

A

by Eq.

Lagrange-Sylvester interpola-

tion polynomial

A

B.

A do

go

A, A Eqs.

p’(A,)

p(A,),

=

(dp(A)/dA)+,

p”(A,), ...,pm*-l( Ai).

8. SPECTRUM O F A MATRIX

Definition. If A is a semisimple operator whose distinct eigenvalues are A,, A,, ..., A,, and i f the values of a function f ( h ) are speciJied for h = h,, A = A,, ..., A = h k , then f (A) is said to be defined on the

spectrum of A.

A A A

158

VI. FUNCTIONS OF A MATRIX

Definition. If A is an operator whose distinct eigenvalues are h i , A, , ..., A, , and mi is the length the longest chain with eigenvalue Xi , then (A) is a function such that for all hi ,the set

have specified values, t h e n f ( A ) is said to be defined on the spectrum of A. (A)

g(A) do

A,

on

(35)

f(A)

f(A),

of on

(A)

A.

Eq. A.

on

A Theorem 3. Given a matrix A whose minimum polynomial is of rank m = Erni, there exists a unique polynomial of rank less than m which assumes a specifid nontrivial set of values on the spectrum of A. This polynomial is the Lagrange-Sylvester interpolation polynomial.

#(A), A.

on

~(h), A on m,

?(A)

+(A) #(A) = X

x(A) T(A)

r(A)

(M4

A

A. ~(h),

+ 44

(36)

r(A) m. on

of A, on

,y(A)y(A).

A.

9.

159

EXAMPLE

r(X)

m {r(h) - $(A)} on ~(h)

s(X),

on

m of m

9. EXAMPLE

B As

R,

Eq. (3).

=

e-jRz

(A

B

C D

) -jflx),

x1

x2

+ B = Ze-iBz C Z + D = e-3Bz

AZ

AZ - B

= ZejBz

CZ - D

= -&Bz

B,

D,

A B

=C

C D

=

-jZsinPz -(j/Z)sinPz

=

COSPZ

=

O S ~ Z

160

VI. FUNCTIONS OF A MATRIX

(32) by

of ?

10. C O MM UTATlV lT Y

A =

A

f(A), go

= f ( A ) ,f

(x) A

A A

A

A. Eqs. (17)

I I I.

A Theorem 4. If A and B are semisimple, then they commute if they have a complete set of eigenvectors in common.

if

and only

10.

161

COMMUTATIVITY

2

any

A

B

4

some

xi, y

on

y =

zaixi

Axi

= hixi

B x ~= pixi

AB

y,

=

BA by

A do S-lAS

S-'BS.

on on

A'

=

B

A',

quasidiag(X,I, , h,I, , ...)

=

S-lAS

(41)

hi.

I,

B' B'

=

S-lBS

=

quasidiag(B, , B, , ...)

Bi

Ic.

A"

B" on T-'A'T

by

A"

=

= (ST)-lA(ST)

B"

= T-'B'T = (ST)-'B(ST)

(42)

162

FUNCTIONS

A MATRIX

=

, T, , ...)

T T

Ti

B,.

Ii , ...)

=

,

=

T,, T,,

...

=

A

T, , T, , ...

A'

A"

(ST)

(43)

...)

(44)

B, , B, , ...

B , , ..., A B

a

A on

A

B

11. F U N C T I O N A L RELATIONS

f(x)

A f(x), f(y).

+y)

1 1.

163

eX+Y

= eXeY

(45)

(x)

on by

x

x

providing

do

Theorem 5.

If A and B commute, then, for any positive integer n

(A

+ B)n = $ (3An-kBk k=O

where n! =

k!(n - k)!

n = 1.

(A

n.

+ BY+' = (A + B) 2 (3 k-0

by

A

(47)

164

k

+1

k,

+

2

"

(k - 1)

An-k+lBk

+ Bn+l

2 (" 1') so(" 1'1

= An+l

+

An-k+lBk

+ Bn+l

k= 1

n+l

=

+ 1,

n n

=

2

A

An+l-kBk

by

B do (46):

Theorem 6.

If A and B commute, then eA+B = eAeB

on on

go

k 0

=

n

n = (r r

+ s), n

go

0

03,

k

13. R

165

NOT CONSTANT

A

(48)

do

(4) 12. DERIVATIVE O F e-jR2, R CONSTANT

(4) x(O),

=

-jRe-iRz

(52)

6.

by

+ O(W)

e-jR*2 = I - jR 6z

O(Gz2)

Gz2

1

= lim -{ -jRGz 82-10

6Z

+ 0(6z2)}e-iR2

- -jRe-iRZ

(53) (4)

(52)

13. R NOT CONSTANT

R x(z) = exp (-j

s'

0

R dz) x(0)

(54)

on

166

+ 6z)

z.

by P(z

+ Sz) = d dz

- ep =

+ sz d P ( z ) + O(622) dz 1 {edP/d2Sz - I} 8 6z

81-0

- p deP p dz

dPjdz

J:R dz

R. z.

R

z,

R X.)

R on z R (54).

14. DERIVATIVE O F A P O W E R

of

z,

14.

DERIVATIVE

167

A POWER

by

U”,

n Theorem 7. If U and may be the same, then

V are two n x n matrix functions of x, which

d dU -dz ( U V ) = - Vdz +U-

dV dz

(57)

derivation.

d -(uv> = dz

+ 62)

1

62-0

V(z U(z V(z

dz

=

+ Gz)V(z + 6z) - U(z)V(z) 6z

+ 6x)

+ Sz) = U(z) + 6z U’(z) + ... + 6z) = V(z) + sz V(z) + ...

d

- (UV) =

U(z

6Z+O

1 {(U + 6z U‘)(V+ 6z V’) - UV) 6z

uv + U’V

d -uu” = u u + UU’ dz

only

2UU’, U

U‘.

(59)

U

n dun - - U’Un-1 + UU‘Un-2 +

dz

... + Un-lU‘

(60)

168

VI. FUNCTIONS OF A MATRIX

n

U‘,

nUn-lU’,

n

by

unu-n = 1 dun dz

+ un-dU-” dz

-u-n

= U’

=0

1 1 +(UU’ + U’U) + -(UW’ + UU’U + U’UZ)+ ... 2! 3!

U’

= U’

t

I

+ u + 21!u2+

15. EXAMPLE

by (54) by

Z 0 = fpdz 0

oz) o/z 0

rRdz=( 0 0

-*/

(62)

169

EXERCISES

R. X(2) =

cos

-IZcos

((-jjz)

B

by

(pz). 2,

on

z,

Exercises

1.

d2x dx u-+b-+cx=O dzL dz - d2x + - - + ( 1 -1- dx )=O n2 dz2 z d z 22 d2x dx 22 - - a(a (c) (9- 1) dz2 dz d2x dz2 (b - h2 COS’ Z)X = 0

+

+

+

d

dx

+

*.

&, (P &) + (4

=0

ddX d2x dx ~ + P ~ + q - dz + Y X = O 2. A

3.

=

(:

1 u)

A

=

=0

(54)

170

VI. FUNCTIONS OF A MATRIX

4.

(I

+ A)-'

=I -A

+ A2 - Ad +

?

* * a

of

B=( 5. =

0,

n x n nilpotent.

idempotent.

= =

I,

involution.

6.

X2+ &BX

+ XB)+ C = 0

?

X2+ 5X

+ 41 = 0

7.

3 = y 2 + *Y dz

+q by

1 du Y=-;&

on dY dz

-Y2+PY+Q ?

8. dM _ -- eZAM, dz

M(0) = I

171

EXERCISES

A

X(z)

9.

z

X(u u

+ v) = X(u)X(v) X(z)

w. --

dz

z,

- Ax@)

A (Hint: separately.) (Comment:

u

Polya's equation.

A

10.

X, = ?

n ?

w

CHAPTER VII

The Matricant

-dx(z) -

dz

-jRx(z)

S = -jR,

_-

dx - sx dz

R

S

z.

R on

z,

-

R

2

x(z) = exp(-jRz)x(O)

(2)

by

1. INTEGRAL MATRIX

by 172

R(z)

1.

a

173

INTEGRAL MATRIX


Rij

Lipschitz Condition. R..(z) is said to obey the Lipschitz condition a? if for any z and z’ in the gzven range, there exists a constant k such that

I R i j ( z )- Rij(z’)I < KI

z - z‘

I

(3)

Rij

R(z)

z

n

n x i ( z ) ( i = 1 , 2, ..., n).

xi(.):

X(z) X(2)

= (XI

x2

XI

.**

x,)

(4)

X(z)

by

X(z)

integral matrix

z.

[

X.

by

X I by

I-

dxkl

dz

dXi j -dz

-

& d dz

...

174

VII. THE MATRICANT

Rki,

IX I

dki

by

i -jl X

IX I

= exp

[- j

dz]

0

=

K

1

I X, I

(7)

Jacobi identity.

X(z)

z.

2. T H E MATRICANT

(5)

on zo

X(zo). X(zo) X(zo) =

X(z)

matricant

M(x, zo). Definition. If R(z) is a system matrix whose coeficients everywhere obey the Lipschitz condition, then zo) is the matricant of the system if zo) =

z,)

2.

175

THE MATRILANT

and 3,) =

I

no of

z.

R(z) zo

zo

x(zo),

z (10)

x ( 4 = M(z, zo)x(.o)

R

Eq. (1 1)

- zo)}

zo) =

Eq.

If M ( z , x ‘ ) is the matricant of a system with system

Theorem 1.

matrix R(z), then z”) =

M(z, z’

(12) x”,

z

M ( z ’ ,x ” ) M ( z , z’)

z‘.

x”).

Lemma 1.

then

If

and Y(z) are two integral matrices of agiven system,

Y ( . )

=

where C is some constant nonsingular matrix. Conversely, if X is an integral matrix, and any nonsingular matrix, then is an integral matrix.

Eq.(5) by

176

VII. THE MATRICANT

X

-

d X d {Y(Y-’X)} dz dz

--

Y

Y

Y

=

-j(RY)Y-lX

=

-1RX

=dY Y-’X

dz d

+Y

d

(Y-’X)

+ Y dz (Y-’X)

d + Y dz - (Y-lX)

X

-jRX. d dz

-(Y-’X) or

Y-’X

C X

=0

=

c

C

Y

C M ( z , z‘)C = M ( z , z”) z”)

z’)

z = z’,

z’)

C

= M(z’, z”)

M ( z ’ ,z) = { M ( z , z‘)}-’

3. MATRICANTS OF RELATED SYSTEMS z’)

(15)

4.

(R, M,(z, z’),

177

PEANO EXPANSION

+ R,).

M,(z, z‘) = M,(z, z’)P(z,z’)

M1

(17)

z = z‘,

P(z’,z’)

(18)

=I

z’)

z, aM1 - - -P aM0

az

az

=

-j(Ro

+ M,

-

-jR&l,P

+ Mo ap

+R P , P (43, + Rl).

M1

M, R, ,

4. P E A N O E X P A N S I O N

(9),

do

(M;’R,M,). R, on R,.

178

VII. THE MATRICANT

by

up Mo = I

dMi - - -jRMi-,

,

dz

0

zo =

M,

= I -j

1' 1' 1'

R ( z ' ) dz'

0

M,

= I -j

R ( z ' ) dz'

-

0

M,

=I

-

j

Mi(0) = I

1'

1 1 11"

2'

dz' R(z')

0

R( z") dz"

0 2'

R(z') dz' -

0

dz' R(z') 0

dz'

S

dz" R(z" )

=

R(z") dz"

0

dz"' R(z"')

-jR,

Mo = I

M,

=I

+ 1'S(z') dz' 0

M,

=I

+

S(z')M,(z') dz' 0

by M(z) = I

+ fS(z') dz' + 0

+

...

S(z') 0

S(z") dz' dz" 0

4.

179

PEANO EXPANSION

Theorem 2. If S ( z ) is continuous and single valued f o r all values of the real variable z, then the series given by Eq. (24) is absolutely and uniformly convergent in anyJinite range of z, and is equal to the matricant with zo = 0.

To

S ( z ) , +(z),

majorant z

z

{I

#(z) =X:?

(25)

S(z).

+(z)

$I(.).

S(z)

n x n

H

H (26)

H2 = nH +(z)

H

, ...

,

= (I

by Eq.

Mii I)

(27)

nonnegative

equal to B (A

< B)

A

A

B i Aii

S

less than or

< Bij < < (Cl(z)H.

(28)

(23),


S(z) dz

0

< 1+

c" # ( 4 H dz

J O

=I+Hs,

180

VII. THE MATRICANT

where

which is a nonnegative real function for all positive z that increases ) 0. monotonically from ~ ( 0 = mod M,


1‘

(mod S)(mod M,) dz

0

< I + I ’ $0 H ( I + T H ) ~ ~

from Eq. (26). Now, integrating by parts, we see that

J

In fact, we have generally

that

Hence,

(32) Suppose, generally, (33)

4.


mod M,,,

181

PEANO EXPANSION

(mod M,)(mod S) dz 0

rfl

i

=I+HC;-z

(34)

i!

i=l

(33)

by

Y

Mi

ij

(33) by

ij

(33).

(33)

Y -+ 00,

mod M


=I

+ -n1 H(ent - 1)

(35)

Mi, (354,

by

(24)

(24) by S

(24),

S

z

(24)

> 0. z < 0,

z.

S(z)

0 z

z

by

182

VII. THE MATRICANT

on

S(z)

star region

by

S(z) 0

z

bound on Corollary 1.

The i j coeficient of the matricant Mij is bounded by

I Mii 1

< Sii + n-1 (enp

11

(36)

on

of

H

$(z)

Corollary 2.

Eq.

Given a nonnegative matrix J such that J2

where

-

01

(37)

= aJ

is a positive real number, and a nonnegative function #(z) such that

the matricant of Eq.

is bounded by

5. ERROR O F APPROXIMATION

on z.

on by

M,(O)

=

I.

bound on

6.

BOUND ON THE DIFFERENCE

So,

183

by

P by P

= Mi'M

by MOP.

P dP- - {Mi1@ - S,)M,,}P,

dz

P(0) = I

(42)

1

P

< I + -n1 H(en"- 1)

(43)

{M;'(S - So)Mo}, {&-l(S - So)Mo}

y

2

P

6. B O U N D ON T H E DIFFERENCE

bound

on bound

dM _ - SM, dz

M(O),= I

_ dN - TN, dz

N(0) = I

-

J2

(44)

= aJ

(45)

184

VII. THE MATRICANT

and nonnegative real functions $ and 7 such that

< #J mod(T - S ) < qJ mod S

Define the integrals =

I’#dz,

= f q d z

K

0

0

for positive z. We also define the partial sums

M,

=I

+ I ’ S ( z ) dz 0

M,+,

=I

+

N,

=I

+ r T ( z ) dz

0

p b1

S(z)M, d z ,

0

modM, We assert that


Jz

(yk-l

D

-

k=l

k

~

~

k

6. This is true for p

=

mod(N,+l - Mp+l)<

BOUND O N THE DIFFERENCE

185

true for p , then from Eq. (51): "I (7J (I

z P

ak-l

+ J , __ k!

by shifting the value of k. Hence, Eq. (53) is established by induction. In the limit, as p -+ co,

- M)< J 2 __ k=l k! ak-l

mod"

{(K

+ v)k-

1 J{ea(K+P) - ap, =a e l 1

= - JeaP{ea" - 1 a

1

(57)

186

VII. THE MATRICANT

bound on of 7. A P P R O X I M A T E E X P A N S I O N

R z.

z by

k z

z

zo).

on z

z,

good good

3 on

As

< 1.

kz on E

-

I

Z

=

+ kz), 1/ Z

k -

kz)/Zo

8.

M,

= esp(-jR,z)

=

(Mi1R1M,) = k/3z

cos /32

( -(j/z,) -j

P=I--(

k

/3z

cos /32

2/32

(-(l/z)cos 2/32

K, P =I-jh/3/’(-;

187

INHOMOGENEOUS EQUATIONS

j s i n 2/32

Eq.

z : cos 2/32) d2 32 2/32

2/32 2/32

jz

z/Z)

jZ(cos a

a - a cos a )

4/3 -( j/Z)(cos ‘Y

+a

Eq.

a

+a a

-

-

a - 1) a cos a )

1

on z Kz.

k.

z,

8. INHOMOGENEOUS EQUATIONS

Eq. dx dz

-- -jRx

y

+ y(z)

z. x,,)

_dM - -jRM, dz

(59)

188

VII. THE MATRICANT

(58),

dM zu

du JU + M -dz = -.jRMu + M - = --J’RMu + y dz

(60),

z = zo ,

uo = x(z,,).

+

K(z, z‘)y(z‘)dz‘

= M(z, zo)x(zo)

20

K(z, z’)= M(z, z0)M-l(z’,zo) =

M(z, zo)M(zo, z’)

=

M(z, z‘)

(63) z’)

9. T H E MULTIPLICATIVE INTEGRAL O F VOLTERRA

multiplicative integral of Volterra.

by R(z) zo zo

z.

< z1 < z2 < ..* < z,

=z

Cauchy

189

9. (12),

1,

M(z, zo)

= M(zn 9

zn-J~Vzn-1,zn-2)

.**

M(z19 zo)

(64)

R(z) M(zi+l ,zi)= e - j R ( z i ) ( d z x )

(65)

A z ~= zi+l - zi

M(z,+, , zi)= I - jR(zi) A z i ~ ( zzo) , =

Axi multiplicative integral

(66)

n

{I - jR(zi) d z i )

(67)

0, { -jR(z)},

--t

zo) =

1'

{I - jR(z) dz}

ZO

R(z)

R(z)

derivative,

by

D,X, dX D,X(z) = --x-1 dz

X(z) R(z):

-ax - - -jRX dz

z

multiplicative

190

VII. THE MATRICANT

by Eq. (13),

D,

1,

of

{I -

dz} = X(z)X-'(z,)

{I -

dz} = dx {X(z)X-l(z,)}X(zo)X-l(z)

d

-

dX(z) X-l(z)

=

-jR(z)

dz

=

-jRX(z)X-l(z) (72)

--jR(x) up

on by

Exercises

1.

of

fi

Z

2.

3. A

by

191

EXERCISES

S

a

Cauchy - a)

=

4.

P'(I

+

csc-1dz)

=

+s

c [])I

c-1

20

C

5.

(b)

+ XD,(Y)X-'

D,(XY)

=

D,(X)

D,(XC)

==

DJ,

D,(CX)

= CD,(X)C-l,

D,(Xt)

= Xt(D,X)tXt-'

C C

(e) DZ(X-l) = -X-'(D,X)X

(f) D,(Xt-l)

=

-(DzX)t

6.

dN

__ =

dz

S?

-NS,

N(0) = I

192

VII. THE MATRICANT

7. Show that the solution of

dX dz

- AX + XB,

X(0)= XO

is given by

x = uxov where dU

-AU,

dV -

-

dx

dz

VB,

U(0) = I V(0) = I

Observe that this is true even if A and B are not constant.

CHAPTER Vlll

Decomposition Theorems and the Jordan Canonical Form up

decomposition decomposition

of

1. DECOMPOSITION

by Definition. A linear vector space S is said to be decomposed into two linear vector spaces S , and S , i f any vector in either S , or S , is also in there exists no vector in S except the null vector that is, simultaneously, a member of both S , and S , , and any vector in S can be expressed as the sum of a vector in S , and a vector in S , . The spaces S , and are said to be a decomposition of S. W e also say that S is the direct sum of S , and S , , and symbolize this relation by

s = s, 0s, 193

(1)

194 by

,

S

, ..., S ,

S, S

x,

,

+ x, ,

xl x

= x,‘

x

x,

S,

+ xi,

xl‘E s, , x2’E s,

(2)’

+ (x,

(3)

0 = (x, - XI’)

-

x,’)

(x, - xl’)= -(x 2’ -x2)

(xl- x,’) E S ,

(x,’ - x,) (x, - x,’)

(4)

. (x,’ - x,) S,

u

x

+

= x,‘

x2’

xl’= x, + u x2)= x2 - u

x by

x

uniquely x

E

= x,

“is

a

+

x2,

x, E s, , xp

s,

(8)

195

1.

Definition. W e say that the operator A decomposes the space S into S , and S , if S , and S , are a decomposition of S , and if they m e both invariant for A.

A x,, ..., xk A

x,, ..., x, on

xk,.,, ..., x,

A 0 A=(OIA,)

k x k

A,

(9)

( n - k x ( n - k).

A,

k

A

(k +

on

on by

S, ,

A

reduce

A. S

=

@

S, : x, , ..., xk ,

A xk+,, ..., x,

.

by

on

A

A,

k x k

A

on B

n,)

A, ( n - k) x ( n - k)

B

196

DECOMPOSITION THEOREMS

all

2. D E C O M P O S I T I O N INTO EIGENSUBSPACES on

S by A

A. A polynomial p ( x ) is said to annihilate a subspace S, = 0 . The polynomial of lowest degree that annihilates S, is called the minimum polynomial of A over S , . Definition.

for A $, for any vector x in S, ,p ( A ) x

I . If A has the minimum polynomial q ( A ) which is factorable into coprime polynomials t,hl(A), Theorem

(11)

V(h) = + 1 ( 4 + 2 ( 4

then A decomposes the whole space S into S, and S , :

s = s, 0s,

(12)

such that S , has the minimum polynomial +,(A) and S ,

+,

, K~(A)

K,(A)

+

Kl(X)+l(4

K2(4+2(4

=1

h by A:

+

kl(A)+l(A) K,(A)+Z(A) = 1 x

(13)

S +l(A)KlWX

+ +,(A)K,(A)X

=x

(14)

2.

197

x x = x1

+ x,

x,

x

(16)

x, . +l(A)XI = +l(A)+,(A)K,(A)X

= dA)K,(A)X = 0

(17)

+2(A)X2

= dA)Kl(A)X = 0

(18)

=

+,(A)+l(A)K,(A)X

+,(A)

so

.

+,(A)

S,

S, .

xo +,(A)%

xo

= +z(A)xo =

0

(19)

x

A xo = Kl(A)+l(A)XO

+ KZ(A)+Z(A)XO = 0

(20)

xo by A.

A,

S,

K,(A)

Ax, Ax,

A. +,(A)

A,

A.

(1

S,

so

.

+, by

. +,(A)

I+,

+,

.

by A.

I+,

O,(X)

S, :

x,

B,(A)x,

=0

x = x1

+ x,

(22)

198

x2

8,(A)

A,

A.

S,

(Clz(A)O,(A)x = 0,

all x

(25)

ES

of q ( A )

+,(A)

S, .

Corollary 1 . If the minimum polynomial set of coprime polynomials

dx> = (Cll(A)(C12(x)

***

of A is factorable into a

(26)

(Clk(’)

then S is decomposable into invariant subspaces s = s, 0s, @ ‘.’ 0s k is the minimum polynomial of

such that

.

Theorem 2. Given a subspace , which may be the whole space, that is invariant with respect to A, and which has the minimum polynomial there exists in Sia vector whose minimum polynomial is vi(A).[The minimum polynomial of a vector x with respect to A is the polynomial of lowest degree such that v(A)x = 0.1

Si (28)

vi(h) = { O i ( A ) } k i irreducible polynomial

F. 71 (A

-

hi). do

on

irreducible polynomial

(A

-

hi).

2.

199

x1 , xg , ..., xk

Si y

Si y =caixi

by Oi(A).

by yi(A)

by no

Oi(A).

by Oi(A) 1s

Lemma. If the minimum polynomials of two vectors x1 and x2 are the coprime polynomials +,(A), $I~(A), then the minimum polynomial of x = x1 x2 is +l(h)+Z(A).

+

x

A

by (+,+J

x:

K(A) K(A)x = K(A)x,

+ K(A)x, = 0 +,(A)

by $J~(A) A,

K(A)

x,.

+,(A)K(A) +,(A)

+g

K(A)

+,(A). .(A)

t,h2(A).

x

(+,I,Q.

x,

+,+,

200

Si,

xi

1

eigensubspaces.

2

do

3. CONGRUENCE A N D FACTOR SPACE Definition. T w o vectors x and y are said to be congruent modulo 3, where S is a subspace, if their difference is a vector in S. W e write

(34)

(35)

z Definition. Given a vector xo and a subspace S , we call the totality of vectors x such that

x = xo

a class or, where we want to be unambiguous, a class modulo S .

(36)

3.

20 1

CONGRUENCE A N D FACTOR SPACE

2.

x

= xo

S

(37)

x=xo+z, ax = ax0

ax z

axo

+ az =

(38)

ax0

+ z',

2'

s

S

(39) (40)

(41)

(ax)^ = a f

by ( )A of x

3

xo

x=xo+u,

S,

3

S

yo

y=yo+v,

x.+ Y = (xo

(u

y

U,VES

+ Yo) + (u + v)

(42)

(43)

(44)

+ v) (45) (46)

Definition. The totality of all classes modulo So is called a factor space of the whole space. If the whole space is S , we sometimes designate the factor space as (SjS,) or call it the factor space modulo So(sometimes called the quotient space.)

202

VIII. DECOMPOSITION THEOREMS

n,

S , x l ,x2,..., x k . x,+~,..., x, f,,, , ..., f,

So (n - k)

k,

So n - k. by

So. z

E

So,

Az.

= xomod S,,

x

(47)

x=xo+z,

AX = AX,

(48)

+ AZ = Axo+ Z'

'49)

So

z'

Ax = Ax,, mod

(50)

A2 = (AX)^

(51)

A on A &i

on x.

f

A,

2, no So

A, $'(A) +'(A)%= 0

(52)

f, +'(A)x = 0 mod

(53)

4. CYCLIC

203

SUBSPACES

$’(A)

x.

A. $‘(A)

x $(A)

So, $’(A),

x, $(A). 4. CYCLIC SUBSPACES

by A.

do

A.

S,

x

by x, Ax, A2x, ..., Ap-lx,

...

(54)

of

on

APx APx

- a,-,Ax -

= --a+

Ap+’x = -a,Ax - -a,Ax

-

-al( -a,x

(55)

- a,A”-lx

a,-,A2x - *.- - a,Apx a,-,A2x - - a,Ap-’x -

a,-,Ax -

A

(56)

- a,Ap-’x)

on

x

x, ..., Av-lx. x, Ax,

. * a ,

Ap-lx.

cyclic subspace generated by x. 6. by

x, Ax,

a * . ,

x,

p.

$(A)

Ap-lx

p

x, x.

$

x, Ax, *.., Ap-lx

S, .

204

VIII. DECOMPOSITION THEOREMS

S, y y

(57)

= X(A)x

( p - l),

X(X)

x

+(A)

x,

by x.

5. DECOMPOSITION INTO CYCLIC SUBSPACES

Theorem 3. The whole space can be decomposed into cyclic subspaces with progressively simpler minimum polynomials.

+,(A).

xl, A, +,(A):

by x1 ,

p

&(A). = n,

p

n

p

n. do

,S/S, .

n

+z(A)

-p .

+,(A).

u a,h2(A)u = 0 mod S ,

&

(60)

#,, +,(A)

=

$u94Y

(61)

5.

205

2,

ci

#,(A)

u u

#,(A)u S, ,

=

#,(A)u # 0,

.

u

#,

(57),

x1 ,

A

no

.

u

0,

S,

S,

S, S, .

P

#,(A)u

$,(A)u = P(A)Xl

by K(A),

(62)

by

K(A)P(A)x, = K(A)+~(A)u = $1(A)u = 0

#,(A)

by

(63)

x1

#,.

#, ,

KP

= 4A)$z(A)Q(A)

(64)

#,

Q(A)x,

P = $28

(65)

+z(A)b - Q(A)xi) = 0

(66)

S, ~2

u (67)

= u - Q(A)x,

x,

by

#, ,

#, by q, .

by

x2

#,

x, , x2 = u

S,

#, , q2.

206 u

S,

x, .

+, .

,

S,

+, by

,Q I

y,.

+2,

$,

x, . by x, :

S, S,:

x , , A x , , A2x,, ..., A+-'x,

(65)

+, .

q

S,

co ,

+ c,Ax, + + cQ-,A"--'xZ

coxz

U(A)

+,

.

+ clA + + Ca-lAq-1)X2

=

(cJ

=

U(A)x,

S, , S,

no

..., cq--l

EE

S,

0

(69)

q

-

1.

x, ,

q.

S,

+ q = n, p + q < n,

p

S, @ S ,

:

Theorem 3'. Given a matrix A the whole space S can be decomposed into cyclic subspaces S , , S , , ..., S , with minimum polynomials #J,(A),+~(A), ..., &&(A), where +,(A) is the minimum polynomial of A, and +i(A) is a divisor of +i-l(')*

Corollary I . S is cyclic (with respect to A) if and only if the degree of the minimum polynomial of A is the same as the dimension of

S, $,(A)

n.

4.

207

CYCLIC SUBSPACES

x

S,

of x

n.

A

n,

n.

Corollary 2 . A cyclic space can be decomposed only into subspaces that are cyclic and whose minimum polynomials are coprime.

+(A)

n

m.

S

by m=n

S

S,

:

s = s, 0s2 S, +,(A)

S,

n2 ,

n,

of

+,(A),

m 2,

m,

S,

S,

+ < (n, +

(m,

m2)

.2)

+,

+(A) m

< m, + m2

+, m

m

=

(73)

= 12

d

+, , (74)

+,

+ m,) d (n, + n,)

(m,

=n

(75)

n. m, = n,

(76)

mz = n2

(77)

m = m,

+ m2

(78)

208

VIII. DECOMPOSITION THEOREMS

Corollary 3. S is indecomposable with respect to A i f it is cyclic and its minimum polynomial is a power of an irreducible polynomial.

Corollary 4. If S is indecomposable with respect to A, then it is cyclic and its minimum polynomial is a power of an irreducible polynomial.

3 1

Theorem 4. The whole space S can always be decomposed into subspaces S , , S , , ..., S, which are cyclic invariant subspaces with respect to A,

each subspace having a minimum polynomial which is a power of an irreducible polynomial. No further decomposition is possible.

S by

3 S

A go

A. no

6. THE JORDAN CANONICAL FORM

(A - hi). 4

A,

S

6.

209

THE JORDAN CANONICAL FORM

( A - hi)k*.

s = S,@S,@-..S, Ax E S , (A - AtI)A~x, =0

(79)

if x E S, if x, E S,

Si

xi

Si

by

S,: (x, , A x , , ..., Ak-'x,)

u1 = (A - AII)kl-l~l ti2 (A - A11)'1-2~1 1

...

Ukl = x1 tikl+1

= (A

-

u1 , ..., uk, u/c,+l . . * I Also, 3

XzI)k2-1~z

S, ,

uk,+kl

(A - A1I)ul = #1(A)x1 = 0 (A - A,I)uZ = (A - AiI)kl-l~l= tll

u1, uz , ..., u, w

S, .

(82)

210

VIII. DECOMPOSITION THEOREMS

A

A,

k, x k, , A, , ,

k, x k, ,

Ai

4

k, , k, ,

no

on

(79)

(A

-

A,)k,

x, (A

-

A,)k.

. S.

no

7.

INVARIANT POLYNOMIALS A N D ELEMENTARY DIVISORS

2 11

by by

111. 7. INVARIANT POLYNOMIALS A N D ELEMENTARY DIVISORS

A.

(A (A (n -

by do

by A.

by

of

A

D,(A).

(A -

Dn(h), Dn-l(X)*

Sl(A), Do(X)

=

1

(87)

D,(A),

1.

(A D,

by D,-l. on -

by

-

by

DkPl D,

.

...

{p,(A)}

A.

212

VIII. DECOMPOSITION THEOREMS

invariant polynomials A.

D,(X)

A.

n

H p , ( h ) = DJX) = 1 A - XI i=l

{p,(A)}

A.

{p,(X)} by invariant.

elementary divisors A.) (A

-

A.

pi(X).

XJk,

3' Theorem 3".

If A i s , a linear operator on the vector space S , then S

can be decomposed:

s = s, @ sz @ ." Sk

(91)

where Si is cyclic invariant subspace with the minimum polynomial +,(A), and where the set +,(A) coincides with the set of invariant polynomials of A. (other than

4 Theorem 4 . If A is a linear operator on the vector space S , and if S is decomposed in any manner into indecomposable invariant subspaces {S,} with minimum polynomials +,(A), then the set {+,(A)}is the set of elementary divisors of A.

213

EXERCISES

by

hi do

on

4’.

3“ 8. CONCLUSIONS

if

Exercises

1.

S, 1, 1, 0, do S , S,: S,:

1, 0, 0, 1, 1, 1,0, 1, 0,

S, 2.

-

S,

S, 0, 1, 0, 0, 2,0,

S , do n x n

A

214

VIII. DECOMPOSITION THEOREMS

S,

3.

S,

A. S,

S,

.

S, S,

S,

A.

S,

4. /O

+j

1

A ?

5. 0 el

=

,

e3 =

(a)

el mod e2

el mod(% e3) (c) col(1, 0, 1) mod e, (d) col(l,O, 1) mod el (f)

6.

by

col(l,O, 1) mod(e, , e3) col(l,O, 1) mod(e,, e,)

1

k) 0

e2 =

CHAPTER I X

The Improper Inner Product K

11,

5,

11, on

E

-

I by

s =

&!?*I

+ El")

(1)

)

K = -1(0

1 2 1 0

K.

f 4,

K

n

x

+ so 215

216

IX. T H E IMPROPER INNER PRODUCT

do

E,

I, (wo

+ nu,)

up

pump.

couple parametric network

by up.

wk W,

by

K=(w

Cf. J. M. Manley and H.

0)

(4)

Rowe, PYOC. IRE (Inst. Radio Engrs.) 44, 904 (1956).

1.

217

THE IMPROPER INNER PRODUCT

K.

K

& 1/uk.

1. T H E IMPROPER I N N E R P R O D U C T

(x,Y>

(6)

= xtKy

K x, x. (X,Y) = (Y,X)*. ( C ) (x, my Bz> = 4x9 Y>

(A)

(x, x)

+

+ B<x,z>

K by

(x, x),

x

K

K

f1

yi on (7)

Y = Z*iYi

K. ai

on

Eq. (8) yi .

218 by

by

s

no (1)

s

x

xtKx

=

0,

K

2. FAILURE O F CAUCHY-SCHWARTZ A N D TRIANGLE I NEQUALlTlES

by

(x, x>

= (YI

Y>

=0

I(%

(X,Y> = 1 y>I2

< (x, x> (Y,Y>

(12)

(13)

3.

219

ORTHOGONAL SETS OF VECTORS

Ah, (x+y,x+y) = 2

{(x

< (x, x)1/2+ (y, y)ll2

+ y, x + Y ) } ” ~

Also,

(15)

Eq.

I1 do do Definition. The vectors x and y are said to be orthogonal under the K-inner-product, or K-orthogonal, ;f

(x, y) = x+Ky = 0

(16)

by

hyperbolic, elliptic. on

3. O R T H O G O N A L SETS O F V E C T O R S

of

by

xi :

y y

=

2 aixi

(17)

of n

ai .

xi ai

If, (1

.

220

IX.

xi,

(x-j,

(18)

Xi) = or.&.. t 23

aij

aj

-j, x,~ (x- j , Y>

=

2 ai<x-j , xi>

aj = ( X - j ,

= apj

(19)

(20)

y)/aj

xi ,

y

up

4. PAIRWISE ORTHOGONALITY

K

( x i , xi)

=

xitKxi

xj ,

0. xi

K

=

(xj, xi)

= xjtKxi

Kxi

#0

(21)

xi. xi (xj , xi)

j

(-21,

4.

PAIRWISE ORTHOGONALITY

i.” j.”

“i

-

i

j,

i

i

-

22 1

j,

i. 217), ( x i ,xi) # 0.

by

Definition. The complete set of vectors ui is pairwise K-orthogonal, i j to each value of the index i there corresponds a unique value (-21 such that

(u-, , u,)

=U

Lp,

(22)

= $-jz,

where the ai are nonzero, possibly complex, numbers.

(4) of

i.

K

a’s a’s

K ai

i

-

a-1

- (x, ,x-,) = (x-, , x,)*

i, ai

(23)

= a,*

i - j # i.

a

a’s

y,

, x,)1/2

= XJX,

(Yj Yi) = Y3tKYi 7

=0

=X J a y

xjtKxi = (aj*)1/2(ai)1/2

-i

if

= ai/(lai 12)1/z

# i, j - i , (yi , yi) = 1.

(1

i

ai /2)1/2

-

(Y -i =

ai

-ai

-

i

(24)

ai

,

(yi , yi)

=

yi

do

ui =

i,

if j

>

Yj>

= YtiKYi = Qij

(25)

222

IX. THE IMPROPER INNER PRODUCT

maximally normalized,

yi

of

K

Eq.

x,tKx, = 2 xZ~KX,== -2 xltKx, = x2tKx1 = 0

1

of

2

N

2.

self-conjugate. 1 Y1

1

=

1

1

Y2 =

9

yltKyl = o1 = 1 yztKy2 == ~2 = -1 YltKY2

= YZtKYl = 0

of u 2 .

OL,

on 1

x1=

(j ) ?

1 x2

=

xltKx,

= x,~Kx, = 0

x,tKx, x,tKxl

= =

= a:

2j

= aI2

1 of

cross-conjugate.

-

2.

-

5.

223

ADJOINT OPERATOR

by

vi -yi

by

by do

+ )pi).

(25), y,

y,

u,tKu,

= u,tKu, = 0

u,tKu,

= u,+Ku, =

1

5. ADJOINT OPERATOR

do do Definition.

The matrix A# is the K-adjoint of A if, for all x and y

in the space (x, AY> =

Y>

(26), xtKAy

= (A#x)tKy = xtA#tKy

x

K.

y,

K

(26)

224

IX. T H E IMPROPER INNER PRODUCT

if

A

A

Definition.

AA# = A#A

AK-lAtK

=

if

A

Definition.

A = A#

KA

=

K-'AtK

= AtK

if

of

A

=

-A#

KA

=

-AtK

=

-K-lAtK

if

A

Definition.

(28)

K-lAtKA

of

by

X

A A =

1 X l2

b

0 (b

b =

1

b' a)

225

6.

A A by 6. O R T H O G O N A L I T Y A N D N O R M A L I T Y

by

A A

by

by

Theorem 1 . If A is K-normal and semisimple with discrete eigenvalues, then its eigenvectors are Kvorthogoaal by pairs.

AX, = Xixi A#Axi

= M#x, = hiA#x,

A#xi

A

hi

A#:

xi,

xi (xi, Axi)

=

(A#xj, xi)

= hi(xj , xi) = pj*(xj,

(hi - P j * ) ( X j ,

Xi) =

xi)

0

i

Eq. hi

(hi - pi*) = 0. (xj , xi) = 0

i i

i, i,

(xi, xi) = 0. xi

( x i ,xi) # 0, -j.

A

A#,

226

IX. THE IMPROPER INNER PRODUCT

A#

by A by =

p,i

xi*

(33)

Theorem 2 . If A is semisimple, and if it has a complete set of eigenvectors that are K-orthogonal, then A is K-normal.

xi

A.

by

by by 4. y i = A#xi - XZixi

(34)

yi yi -j, (x,

j>

yi)

=

(Ax,

==

(A2j -

j

9

xi) -

i<X-

j >

i)(x- j , xi>

i =j , yi

( h L j - h*,J i #j.

(xej, xi)

xi xi

xi)

yi

A#: A#xi = XCixi

A AA#xi

= :A

iAxi

= hi/\*

A#,

ixi

A#Axi = h,A#x., = XiA*,,xi

AA# = A#A

A 2

on

A

(35)

7.

227

K - H E R M I T I A N MATRICES

1

A by

by

A A#, A

of m,

k ( m - k)

m.

Theorem 3. If A is K-normal, then its eigenvectors and generalized eigenvectors can be chosen so that they are pairwise K-orthogonal.

2

on

7. K-HERMITIAN MATRICES

A on

A

?

K?

so,

K.

so,

Theorem 4. The eigenvalues of a K-hermitian matrix, where K may be indefinite, are either real or else occur in complex conjugate pairs. If the

228

IX. T H E IMPROPER INNER PRODUCT

matrix is not semisimple, the conjugacy relations must relate vectors in a chain or in chains of the same length in the opposite sequence of rank.

xi A, by AX, = hixi ei

0

1

+

(36)

~ i ~ i - 1

xi

on

A, (xi, AX^) = (Axj , xi) U x j xi) 9

+ ‘i(xj

9

xi-1) = Xj*(xj xi> 9

-i,

j

+ ~j(xj-1,xi)

(37)

on

xi

xi

i

-

=

AZi

(38)

hi hi,

hi* j

=

-(i

-

, Xi-&

‘,<X-(,-l)

ei =

1,

= L(i-l)(X-(i-l)-l

*

Xi>

xi

-i

i. on on

Theorem 5. If the eigenvalues of A are either real or else occur in conjugate pairs such that chains of the same length are associated with each of the complex conjugate pairs of eigenvalues, then there exists a K for which A is K-hermitian.

7.

229

K-HERMITIAN MATRICES

by

KA

(40)

= AtK

by

V, K'A'

= (StKS)(S-'AS) =

StKAS

= StAtKS

(42)

= (StA+St-')(S+KS) = A t K '

A' A

A

=

At

=

...)

mi x mi

J$Lhi*)

(43)

..a)

(44) hi*:

(44)

S do K

do

K,, , ...)

=

mi x mi

JLi

(46)

230

IX. THE IMPROPER INNER PRODUCT

K so

K

&,

by

A

K

mi = 1 so

A A,,

Jv,JAl), Jnb,(h2),so

A,*. A,

m, = ml = m

=

A,*.

K

=

(

:::j

0 K0 m, Kml

...

K,

by on A

K A

K

K on

K so on

so

5

A

K.

8. K-UNITARY MATRICES

4, Theorem 6. The eigenvalues of a K-unitary matrix are either of unit magnitude or else occur in pairs such that one is the complex conjugate of the reciprocal of the other:

AiXEj

=

1

(49)

If the matrix is not semisimple, the conjugacy relations relate vectors in a chain, or in chains of the same length, in opposite sequence of rank. by

on

5

4.

8.

23 1

K-UNITARY MATRICES

Theorem 7. If the eigenvalues of a matrix A are either of unit magnitude or else occur in pairs satisjying and such that chains of the same length can be conjugated, then there exists a K such that A is K-unitary.

5. (46)

AtKA

AtKA

(50)

=K

= (StAtSt-l)(StKS)(S-lAS) z

StAtUS

==

StKS

=K

A' A A

=

(51)

K-lAt-lK

A At-l

A.

0 J;-'(X)

0 0

0 0

= -p4

...

cL

cL3

**.

...

p = 1/X*

pi = X i .

Xi Xei

do

A-l.

do do by

V,

=

pi.

232

IX. THE IMPROPER INNER PRODUCT

i

xi

(JL-'(A)

- pI)xi = J;;l(A).

K

== (XI

v

xz v

--*,

(53)

xm)

JL-l(A)

Jm(p).

K

'

K (50), AtKtA = Kt

K

(49), Ka

Kt.

= eiaK

+ e-jaKt K

a.

Ka

(ejaTi

+

rli, e-jaqi*).

a

K K on

K 9. O R T H O G O N A L I Z A T I O N O F A SET O F VECTORS

K

xi

111,

wi w.tx. a 3 = 6.. a3

(54)

10.

23 3

K

RANGE OF

K xLiKxj

(55)

= ui6,j

xTiK

=u , ~ !

Kx,~

=~

i

~

i

of

w-~ W t k

x-~.

.

x-~

of

UkWk,

K. 17, of

of

A

At, of

10. RANGE O F K

K's by Theorem 8. If A is K-normal, K-hermitian, or K-unitary, and S is any nonsingular matrix which commutes with A, then A is also K'-normal, K'-hermitian, or K'-unitary, respectively, with

K

= KS

providing S is such that K' is hermitian.

(58)

234

IX. THE IMPROPER INNER PRODUCT

A

S,

S

S-'A

=

A,

S-I:

AS-'

so

AK-lAtK

= K-'A+KA

A(KS)p'At(KS) = AS-lK-lAtKS = S-'(AK-lAtK)S = S-'(K-'AtKA)S = (KS)-'At(KS)A

A

A so

KA

= AtK

(KS)A = KAS

A A

= At(KS)

so

AtKA AtKSA

A f(A)

=K

=

(AtKA)S = K S

A,

K f(x)

K,

K.

A

A

Kf(A), A.

on

a

A. A

on

K's.

on

Kl

=

(;

A)

E

I K-

-

(59)

11.

DETERMINATION

235

A METRIC

2.

Sz =

1 --{I E + zz 12 + 1 E - zz 12} 22

2

K

of 2, 11. D E T E R M I N A T I O N O F A M E T R I C

K on

of

K A (29)

K K

A

K A

236

IX. THE IMPROPER INNER PRODUCT

K

Eq. (4).

by

K, K,

by

a

do

K

a priori a.

by

no

Eq. (4). Eq. no

Eq. by Eqs.

231

EXERCISES

Exercises

K

A

1. 2.

\-1

+j

+

O /

A

/o 1 0 0 0 0 0

A.

A 3. x1 = col(l,O,j , 0) = col(1, 0, - j , 0) x3 = col(0, 1 , 0, j ) x4 = col(0, 1,0, -j) x2

-

1 1, (b) 1 - 1 , (c) 1-2, 1-4,

2 -2, 2-2, 3 -4 2 -3

3 -3, 3-3,

1

4-4,

u1 = u2 = u3 = u,, =

4-4,

u1=u2=+Iru3=u4=-l

4.

K=(q 8

n)

0 0 0 1

238

IX. THE IMPROPER INNER PRODUCT

col(1, 0, 0) -1, 0, - j ) ( c ) col(1, -j, - 1 , j ) col(1, 1, j , - j )

col(0, 1, 0, col(1, 0, - j , j ) col(1, 1,j ) col(j, 1 , 1,j)

K

5.

K

K

=

O P (P 0) 0.

P

K

P. Eqs.

(4).

of

do 6.

6.

CHAPTER X

The Dyad Expansion and Its Application

by

dyad

outer product

1. T H E O U T E R P R O D U C T O F TWO VECTORS

outer product

x xty.

xyt. x

y

1x 1 n x 1

n x 1 1 x n

y

xty, n x 1. xyt, on n x n.

1 x n

x

y. yt x. xyt

y 239

x

240

X. THE

xyt on

u.

(xyt)u = (ytu)x

(1)

ytu y

u, x

y, x

x. y u on y

(xyt)u x y xy' u

h

(xyt)u = xu

(2)

x,

u (n -

x

(ytx).

h y

x

y

y by (n -

2.

(yty)x.

K-DYADS

ui if j not 4 - 0,. = u. j--i

u,tKu, = 0

ui

(3)

=

j,

ui

4,

u,tKuj

K

3.

24 1

IDEMPOTENCY AND NILPOTENCY

K, Eij Eij

= u,u,uL~K

(4)

ui

-j

K (3). E

~

= ~ =

E

~

(5)

~

a,a,(~tj~~s)~,~t,~ (3),

EijEst= 0 =Eit

s

# j s =j

do

3. IDEMPOTENCY A N D NILPOTENCY

(4)

Eii . E:i

=(u~u~u~~K)(u~u~uL~K)

(7)

= aiuiuLiK = Eii

E, by

' A set to

Eii . Eii idempotent,

elements that have the group property but do not

a semigroup.

a gioup are said

242

X. T H E DYAD EXPANSION

i #j, EiiEij= ( u ~ u ~ u L ~ K ) ( u ~ u ~ u =L0~ K )

nilpotent,

f(x) = a,

+ a,x + ... = 2 anxn

(9)

n=O

f(kEii)

+ 2 PEYi = aOI+ {f(k) - a,}Eii

= aOI

i #j, f(kEij) = u,I

(10)

+ a,Eij

4. EXPANSION O F A N ARBITRARY MATRIX

Eij

ui n x n

A i3

by ufSnLK

by u,

i

=

rn

=

. n,

by on

4.

243

EXPANSION OF A N ARBITRARY MATRIX

A, of (aij)

A, ;

A

u,

am, = a,ut,,~h~u,= o =A,

(8),

m #n i f m = n

(16)

A

(am,) A

=

2 AiEii z

u, A,

u,-~ Au,

= XU,

+

amn = unu!imK(Aun

=A

~ n - 1

+

Un-1)

n = m

-

m =n -1

=0

(amn),

(=

A on

U,U~-~

on I =CEii i

by

k

EijEkh AB

=

2 ih

AB of

E,, .

=

(20)

aij6,h) Eih j

of

244

X. THE DYAD EXPANSION

5. F U N C T I O N S O F A M A T R I X

ui

A, f(A),

A, f(x)

A A2 = zXiAjEiiEjj= z X t E i i

(21)

ij

by An = zXinEii i

A,

on by

f(x)

A. 6.

EXAMPLE

on

2

-

R

7.

CONSTANT COEFFICIENT DIFFERENTIAL EQUATIONS

245

u1tKul = 1 u~~KuZ == -1 ultKu2 = u,~Ku, = 0

-

cos pz

(-(j,z)

pz

-jzcossin pz

7. DIFFERENTIAL EQUATIONS WITH CONSTANT CO EFFlCI E NTS

-dx(z) -

dz

-jRx(z),

x(0)

= x0

ui

R:

R

R

=

C~ijEij

(rij)

ui

x(x) X(.)

ui

=

2

on (26)

.i(Z)Ui

R

246

X. T H E DYAD EXPANSION

(24),

=

-j

=

-j

2 ripkuiUiUkjKUk ~ ~ ~ a ~ u ~ u ~ u ~

ui

uk

(27)

k:

R

A,, ...)

rij

z =

0

(26), xo =

Cyi(0)Ui

by ojuLiK, aj(0) = aiuLjKx,

R x(z) =

2 I

ui,

ai(0)

of

e-jAir

aiuLiKxOui

x,,,

8.

247

PERTURBATION THEORY, NONDEGENERATE CASE

up

R

on

up

rij yi

yi

-jhiz)

m

z

m

- i,

i

8. P E R T U R B A T I O N T H E O R Y , N O N D E G E N E R A T E CASE

As

by R

= R,

+ cR1

(35)

perturbation parameter.

E

R, E.

by do

k

up

R, R,

eR1 E.

?

248

X. T H E DYAD EXPANSION ti,

R, u, by

E

K

K

R,,

K

R

E.

K

R,, u, Eq. (3).

hi

R, 5),

on

up‘

Ru,’ = h,’~,’ ti,’

= u,

+

(38)

EV,

(39)

pp.

vp

vp :

tii

v,

R,

=

Eqs. (39)

aeiui

(40)

(38), E

R, . Riu,

E

+ Rev,

+

= P~U,

Xpvp

(42)

8.

249

PERTURBATION THEORY, NONDEGENERATE CASE

Eqs. (36),

Ro

(37)

R1,

Eq.

v,) Eijtlk

=u~u~uL~Ku~

=O

if K # j

K

= uiujui

(43) =j

ui Eq. (44)

+

u i ( u L i ~ ~ l u p )h i a p i = p p a p i

i

+ pa pi

(45)

=p,

p# = aiutiKR,u,

i # p,

(46)

Eq. (45) (47)

api = U i U f i K R 1 U p / ( h u - h i )

vp ,

Eq.

a p p, app

up

R, ,

app

up on

K-or-

250

X. THE DYAD EXPANSION

9. D E G E N E R A T E CASE

by (A,

Eq.

-

hi)

by

R, ,

u2

ul A:

R , u ~= XU, R,u,

= Au,

R,

no of A. A

do

R,

As

K

ui

so may so so a

of

ul

R, . of

E,

E

by ul

u2

u2

9.

25 1

DEGENERATE CASE

uz

ul

R, . w

+ atlz

= ti1

(49)

01

w‘ w’

=w

+

a , a1

wz, v1

w1

aP,

wl’

(50)

EV

wz’

v,

.

(40), (Ro

+

ERl)(U1+

auz

+

EV) =

(A

+

+ auz +

EP)(U~

w

E

R,

(51)

EV)

A.

Rl(u1

+

+ ROV

0 ~ ~ 2 )

= Av

+ ~ ( u+i auz)

(52)

(41)

u, ,

ul

a , ~ t l+~a ~( T l ~1 l~=t l ~ ~ l ~ 2 u Z ~ t+Zaa2ut2KRluz ~ ~ l ~=laP

(54) (55)

a p.

p OL.

(55) of

252

X. THE DYAD EXPANSION

w’s

u2. up

a

a.

uL,KRlu,

v do

w1

wl.

R

(24),

(53),

01

p

B2 - 4AC

= {alUtlKRltil - u~u:&R,u,}~

+

(56)

E,

pl p1

p2

p2

up’

ul’

on

Eq.

R, ,

R1, by

al

a2

by on by

K

10. APPROXIMATE MATRICANT

253

11.

R do

(34),

by

R, + E R ~ ,

R up

R, R

R,

R

K.

R,

by

R1,

E

E,

(34) z, z.

R (34)

k

R

(k - 1)

z

z z

R, R, by R1

R. (34)

z.

by on z, good 11. CONCLUSION

254

X. THE DYAD EXPANSION

(24),

K.

by

no

E.

Ro

E

so

R,

R, K

do

on

Exercises

1.

x;

0 0 0 1

K=(;

XI =

1, 0, j , O),

xg =

1,0, j )

x2 =

1,0, -j,

x4 =

1 , 0, - j )

xl,x2,x3, 1, - 1, j , -j,

x4

255

EXERCISES

Ax,

= jxl

,

Ax, = -jx,

Ax,

= jx,

+ xl,

Ax, = -jx4

+ x3

Ax,

=jxl,

Ax, = -jx,

+ x4

Ax, = jxz

+ x1 ,

Ax, = -jx4

Ax,

= jx,

,

Ax,

= jx,

Ax,

= jx,

+ x, ,

Ax,

=

2.

-jx,

x 51 :).

0 1

Ro=(:

+ x,

0 0

0 0 0 0 0 0 0 0 1 o)

0 0 0 0

0 0 - 1 0 b

R

= R,

+ ER,

R

E

E,

K=(i;8

0 1 0 0

R (a)

a = 5, b

=

u = 3,

=2

b

6

( c ) a = 2, b = I 2

on

E

R,

256

X . T H E D Y A D EXPANSION

by

J . Appl. Phys. 31 2028-2036 3.

n x n

n (Comment:

Eij

n2 (Caution:

3,

4. bv

dX dz

S

1

-

Z-LX

sx,

X(0) = I

a

S K

5.

E

uvtK.

EX = y

x

y

y

u.

by x

= LXV/(V~KV)

a?

a

6.

by (uvt, xyt)

= (v, X X Y , u>

11,

4, 7.

A

K A

=I

+UV~K

AX = y ?

?

257

EXERCISES

(Hint: x

+ U V ~ K X= y

by vtK,

(vtKx).

x.)

k 8.

Eij

K,

(4).

A on

on

aij \

?

C H A P T E R XI

Projectors projectors. of

on

of

on

xy

z

do

on

by

on

P

idempotent

P2 = P).

1. D E F I N I T I O N O F A PROJECTOR

S

S, x

S,.

S

S, 258

decomposed

x, + x, ,

S,

S,

x1

1.

S, S,

.

x,

S,

S,

no disjoint.

of S,

S,

S of S ,

259

DEFINITION OF A PROJECTOR

,

S,

S, ,

S

direct sum

s = s,0s, Definition.

S into S , and S , so that

Given a decomposition x

= XI

+ x, ,

(1)

x1 E s, , x, E s,

(2)

for any x E S , the operator P that carries x into x, is called the projector on S, along S, .

on S,

P

/

3

Y' ,

/

/

1

/

s

/

FIG. 1.

The projection

on S , ,

x

x

y'

A projector is a linear homogeneous operator. x Px

a

y

,

u on s along y and y'

of

S, is Theorem I .

on S, .

u

1

= x, = x,

F

+ x2,

x1 E s,, x, E s,

on

260

XI. PROJECTORS

s, + Yz , Yl s, x + Y = (x1 + + (x2 + P(x + Y) = x1 + y, = px + 4r

Y

= Yl

E

7

Y2 E

31)

Y2)

Px = 0.

x

k x k,

P

on

K 2. IDEMPOTENCY Theorem 2. A linear operator P is a projector idempotent-i. e., P2 = P

on Px

S,

and only

if

it is

x

= x1

x,

S, pzx

.

if

,

(x, + 0).

= PX, = XI = Px

x pz

=p

S,

x, Px,

= x,

(3)

Px2 = 0

(4)

x,

S,

2. S,

.

26 1

IDEMPOTENCY

u

S, ,

Pu

=u

Pu

=0

S,

S, ,

u

S,

S,

x. x

= Px

+ (I - P)x

XI = Px,

x,

x2

= (I - P)x

S,

Px, = P2x x,

= Px = x,

S,

Px, = (P- P”x

x

= x,

+ x, , S,

= Px - Px = 0

XI E

s, ,

x, E s,

S, .

Corollary 1 . If P is the projector on S , along S , , where S = S, @ S, , then P is semisimple with only the eigenvalues 1 and 0, and S, is spanned by the eigenvectors with unit eigenvalue, S, by the eigenvectors with zero eigenvalue.

Eqs. (3) S,

(4)

S,.

Corollary 2. If P is the projector on S, along S, , then (I - P)is the projector on S, along S, .

262

XI. PROJECTORS

3. COMBINATIONS O F PROJECTORS

on

.

on Theorem 3.

(P,

+ P,) is a projector if and only if P,P,

= P,P, = 0

If so, then it is the projector on R,

along

@

n

.

n

intersection of

and .)

+ (P,

+ P,)" = + PIP, + P,P, + P," = P, + P, PI2

P,P,

+ P,P,

=0

(5)

(6)

,

by

P,P,

+ P,P,P,

=0

by P,P,P,

+ P,P, = 0

PIP, - P2P, = 0

(6), PIP, = P,P,

=0

+ P,) (51, (P,

+ P,)

(7) '

+ P,I2

= P,

+ P,

3.

P

=

P,

by P,

263

COMBINATIONS OF PROJECTORS

+ P,

z

P,

+ x, + y2

z = x1 =

x1 E Rl

y11

Rz

XZ

9

,

y1 E

9

Y,

N,

E N2

P, or

z z

Pz

Pz = (P,+ P2)z= P,z + P,z = Pl(X1 + YJ P,(% Yz)

=

= x1

+

+

+

x2

z z

z,

=

.

+ x, , Pz = P,z + PZZ

=

,

R,

x,

+ Yl) + PdX2 + YZ) PIX, + P,xz

= Pl(X1 =

= x1

+

x2

=z

,

z

+ R, .

z

, so P1Z= P,z = z =0 z = P,z = P1Pzz

P,P,

=

R,

0 by R

N, nN, .

z

= Rl

N, N, nN, ,

Pz = P,z by P,

0R2 Pz

=

(P,

+ PZZ = 0

P, ,

P,2z+ P1P,Z= P1Z= 0

+ P2)z = 0.

Pz

=

z

0,

264

XI.

P2P1z+ P,2z= P,z = 0

P,P, = P,P,

=;’

z

0.

N2,

N,

N , nN,

.

(P,- P,) is a projector if and only if

Theorem 4.

P,P, = P,P, = P,

If so, then R

=

R , n N , and N

=

N , 0R, .

- P2j

I - (P,- P,) = (I- P,)+ P, 3,

P,(I - P,) = (I - P,)P, = 0

I

-P =

I

-

(P,- P,) = (I - P,)+ N,

R

.

by

(I - PI),

3, Rl :

= R,nN,

(I (I -

N,

3

by

Nl N

P

Theorem 5.

=

=R,@N,

P,P,is a projector if P,P, = P,P,

If so, then R

=

R, n R , and N

N

=

N,

(9)

+ N, . N,

N,,

do =

P12=

N, do

=

N,,

P, =

,

3.

COMBINATIONS OF PROJECTORS

265

Eq.

Pa= (P,P,)(P,P,)= P,P,2P, = (PIP,)P, = (P,P,)P,= P2P,= P,P, = P

P

Eq. (9) z R,so z = Pz

P,z = P,Pz = P,",Z z

R, .

.

R, ,

R,

if z

P,P,z = z

=

P,z = z = P,z 2 = P,z = P,P,z R

=

=

Pz

R,nH,

N,

z

Pz = P,P,z = 0 N, .

P,z

P,z

.

N,

z z = P2z

+ (I

-

z N, .

(I - Pz)z N, if z

N,

z=

Pz = z

z

.

P,)z N, N

+ Z, , N, , PZPIZ, + P,P2z2= 0 2, E

N. N

=

N,

+N,

P,z

2, E N ,

N , . Also,

266

XI. PROJECTORS

4. INVARIANT SUBSPACES

of

A Theorem 6.

If S , is invariant with respect to A, then PAP

(10)

= AP

.

for every projector onto S,

S,

s = s,@ s, P

S, x = x,

+ x, ,

S.

S, XI

.

s, ,

x2

s,

by Px PAPx

= P(Ax,) = Ax,

A,so

S,

= x1

S,.

Ax,

PAPx

= AX, = APx

x, PAP

Theorem 7.

= AP

If, for some projector P onto a subspace S , along S , , PAP

= AP

then S , is invariant for A.

s = s,@ s, P

S,

S, S, ,

x

x

PAX = PAPx

Ax

S, . a

S,

=

.

Px,

= APx = AX

A.

267

5.

Theorem 8 . If S = S , @ S, , then A decomposes the whole space into the invariant subspaces S, and S , if and only if

PA

= AP

.

where P is the projector onto S, along

by S

x1 Ax,

Ax,

0S, .

=

= APx, = PAX,

S,,

S, ,

x, PAX, = APx,

Ax,

.

S,

S, ,

=0

A

A

by

6

S,

PAP

S2

= AP

6

(I

(I - P)A(I - P) = A(I A - P A - AP + P A P

PA

-

- P):

P)

= A - AP

= AP

5. EIGENSUBSPACES

x

SAA

(A - h I ) k ~ =0

k

eigensubspace of

by A.

x

SAA,

(1 1)

A.

k

A (Ax).

A.

268

XI. PROJECTORS

of

of

no

of

on

Pi

of

ZAjitA, @ Sy. of Pi

SP

on

by

hi.

of

by of

Pipj = 0

3, (Pi SAi

i# j

(14

+ Pi)

SAi.

&Pi Z P I=I *

R

of

6. SEMISIMPLE M A T R I C E S

A A

=

chipi

xi

hi ,

Axj = ZXiPixj i

We

{Pi}

on

Eq.

=h

j ~ j

6.

269

SEMISIMPLE MATRICES

by An = z A i n P i .7

f(x) A), f(A)

=

zf(hi)Pi

(15)

z

d ~ M ( z= ) -jRM,

M(0) = I

R

(14) on

(16) on

Pi, z.

R z. M(Z) = x m i ( z ) P i

(17)

3) for all i

mi(0) = 1

Pi

=

-j C h i m i p i i

by P, ,

(18) m,

=

=

2

I

hi dzl

-j 0

1

hi dzl P i

-j 0

(18)

270

XI. PROJECTORS

7. N O N S E M I S I M P L E M A T R I C E S

A

A on

Pi by

Hi

Hi

hi,

hi, by

hi . A

=

z(hiPi

+ Hi)

z

Hi

Pi

Pi2 = P i Pipj = 0, #j PiHj = HjPi = 0, i # .j PiHi == Hi = Hipi Hik = 0, k 3 ki

ki

hi.

A'

+ H&P, + HJ = 2 (X,2Pi + 2hiHi + H:)

=

z(X,Pi ij

2

by

H$i = 0. f(x)

hi. off(x),

8.

27 1

DETERMINATION OF P , , A SEMISIMPLE

8. D E T E R M I N A T I O N OF P i , A SEMISIMPLE

A

P,

on

~ ( h ) A.

A

?(A)

A,

.

A,

O,(A) O,,(A)

by i#n

OJA)

P, :

x,

P,

h, # A,,

Pnx, = x,.

P,x, x,

A on

by

on

A on A,,

272

XI. PROJECTORS

9. A NOT SEMISIMPLE

A

Pi,

by

A = (A - Al)2(A

A

= hipi

- A,)2

+ 4P2 + HI + H,

as

A - A11 = (A2 - A,)P, A - A21 = (A1 - A,)P1

+ HI + H,

+ HI + H2 ,

by HI H2

= (A - A1I)Pl = (A - A,I)P2

H12 = H2, = 0,

+ 2(A2 - A1)HZ - A,)”P, + 2(A1 - AZ)H1

(A - &I)’’ = (A2 - A1)2P,

(A - 41)” = (A,

H, (A (A -

*(A,

(28),

= 2(A2 - Al){A - *(Al

* 2 ( 4 - &){A - *(A1

+ h2)

+ A2)I}P, + A,)I}P, {A - +(A,

A,

(28),

(29)

+ X,)I}

H, . by on

Hi by

10.

273

10. T H E RESOLVANT

A

F(X) = (XI - A)-'

(30)

resolvant.

F(A) A.

$

1 277j

- F(A) dA

F(A), F(h) F(h),

Eq.

A

(A - hi)-'Pi

F(X) =

A

Eq.

F(X) =

/(A

-

&-'Pi + (A

- hi)-2Hi

+ (A - hi)-3H: + ...

a

=

'f

P.- -. ' - 277J

a

hi

I

Pi.

F(A) dA

hi.

+ P,).

Hi

Eq. Hi

= (A - XJ)Pi

Xi)(XI - A)-, dh

(32)

(33)

274

XI. PROJECTORS

Pi

F(h).

on

11. ORTHOGO NALlTY

1. 1,

u on x

x,

(x, y )

on x.

(x, y’), xl’

K

by x,’ x.

( u - x,’)

on 1, -

t), 12

I,

= x2 - t 2

by u, ,

(3by u, ,S, S,

S, ,

1 1.

275

ORTHOGONALITY

S, . S, .

S, on

S, S, :

P,P,

=

0.

of

by K-adjoint projectors, Pi#,

by pi# = K-lPitK

(34)

Pi# Hi# i

Pi.

Hi :

A

(35)

= K-'HitK

K (4) Pi# = P-i Hi# hi*

A A#

= =

z

(hipi

= H-i =

+ Hi)# =

z(h-iP-i

(36)

-2

+X i )

z

(hi*Pi#

+ Hi#)

=A

(34) Pf-2 =

Pi#.

Pi#Pj# = K-l(P,P,)tK

(34)

(35)

=0

i#j

276

XI. PROJECTORS

A Eq.

12. CONCLUSIONS

X

on

Exercises

M

1.

M

N,

1, 1,

1, - 1,

N

1 , 0,

N

M 1,

0,

M 0,

1, 1, j , 1, j , j ) ; 1, j , 0 ) ; 1, 0, j , -j).

M

2.

A

u

N

2,

v

=I

+U V ~ K

N

277

EXERCISES

A

A

k

I

A

+ ulv1tK + + u ~ v ~ + K

=I

vitKuj = 0 i #j.

7.)

X, 3.

P

Q A

= PAP

=

I

- P,

+ PAQ + QAP + QAQ

A B B’ AA+BC’ (C D)(C’ D’) = C A DC‘

(

4.

P,

+

AB’+BD‘ CB’ DD’

+

P,

(PI+ Pz-PIP,)

U

5.

)

involution

U2= I.

U=2P-I n 6.

x

n

x

n

n

P

P

P 5)

(2P - I) ? 7.

?

?

5.)

8.

(I + UP)

P (I u =

+ P)-1 = I - OLP 1+a ?

by

278 9.

XI. PROJECTORS

P,, P, , PI

+ P, + P,

=1

PIP,=P2P,=PIP,=P,P, =P,P, =P,P,

=0

CHAPTER XI1

Singular and Rectangular Operators u,

v, u

= TV

(1)

T T

by

T-l.

T

no

u u u

,;

2,

,; +

+

+

u = a X v

(i x )

”v

.;

“a V,

2.

“v

”u

i.

U, V

= j[RU] = j(RU

V

-

R

UR)

U.’

T = j[R, 1 We shall use square brackets, [

, 1,

[A, B]

AB

219

-

n x n (4)

to denote the commutator 1

(3)

BA

280

XII. SINGULAR A N D RECTANGULAR OPERATORS

V

U

R

R, A

R,

dR -- = j [ R ,A] dz

A

(5)

R(z) (5) of x .

A A

(5),

[R,1.

dR/dz (1) n x m, n < m,

m

n

u v. m,

u

m-

on

n

> m,

on

v,

1. ABSTRACT FORMULATION

by Definition. The domain S , of an operator such that the operation of T is defined.

T is the subspace of vectors

1.

28 1

ABSTRACT FORMULATION

do

no

do by

u. Definition. The range S, of an operator T is the subspace spanned by the vectors obtained by T operating on any vector in its domain. W e can express this formally by writing

or by saying that the range of

T is T operating on its domain.

T

by

Definition. The null space S , of an operator T is the subspace of its domain such that T operating on any vector in S , vanishes:

Tx = O

if xisin S ,

T

T

by

Definition. A set of h e a d y independent vectors x, , ..., xk are called progenitors of the range of T if the set Tx,, Tx,, ..., Tx,form a basis for the range of T.

Tx,,..., Tx, (a)

k

k

x, , ..., xk c, -&xi

T Tx,

=0

(7)

282

XII. SINGULAR A N D RECTANGULAR OPERATORS

on

u

v.

u,

by

C,

v, S,. 2. SEMISIMPLE T

xi

S

xi

S., S,

S,

Pi on

do

rn

T

=

CXiPi

(9)

i-1

hi .

m

u

=

TV = ~ & P , V i=l

2.

SEMISIMPLE

283

T

Pk

by

k

Eq.

v

Eq. m

v

=

Z+PiU

+ vg

i=l

v,

m

= 2 ( l/hi)Piu,

(11')

S,

i=l

of Eq.

(I - P,)

P, S, 8,

Eq.

V(h) = h

on

1)

n'

(A -

hi . V(X) = h{l - q x ) }

O(h)

(13)

~(h).

284

X I I . SINGULAR A N D RECTA.NGULAR OPERATORS

q(T) = 0 T

= T{I

-

TB(T)}

= O(T)T'

T

O(T). O(T)T {O(T)T}' = B(T){B(T)T'} = B(T)T

T,

on -

O(T)T}

(15)

T.

on

v

T

x:

on u = TV = T'x

(16)

by O(T)

(14),

B(T)u = B(T)T%= TX = v

(17)

(I).

T, by

v = B(T)u y

+ {I - B(T)T}y

u

v 3. EXAMPLE

T

by

-3

a

x

-I

ff

4

=a

x

-I

-3

-

3

+

*

on (18)

T

+

ZyPi.

+ + +

(a x v ) = a(a * v ) - v(a a)

3.

285

EXAMPLE

2,

% v = --a

+

+

v

- a)

+ + +

x

./(-a

= --a

+

x

= --a

+

x

+ ./-a2

+ k-a+

+

./a2

k

0

so

-a,

-av

(z x )

on T

=

(

0

--a,

+-a9 =0 -a2

= -ax2

_-

'

-a,)

0

a,

Ta

A3

-av

0

a, --av

a2

=

+

0.

-ay2

+

T3=T

0 1

B(T) = - -T -a2

(2 x ).

1 (:

2 86

XII. SINGULAR A N D RECTANGULAR OPERATORS

(3) 4. NOT SEMISIMPLE

may

STo,

3), h c p

Eq. by by

on

S,

S,

S,

proper

by

T#:

T#,

T#u = T#Tv

(Th) (T#T)

S

=

0 x

T

(19)

u

(T#T)# = T#T

(20)

s = sys)

(21)

T#T, Sx,

v by S ,

4. v

287

= O(T#T)T#u, T.

u

on

T,

mod S ,

T#u

Definition. Given an inner product relation and a subspace S, , then S , is the orthogonal complement of S, iJ for any x in S, , and any y in S, (x,Y> = 0

Theorem. If the given inner product relation is a proper one, then a subspace S, and its orthogonal complement S , is a decomposition of the whole space.

. S, no

S,

u

,u, ,

v: u=u,+v

(v, v)

= (u - U 1 , U - Ul)

ul u on

, by

- u,,)

u, . u,

= uo

+ ciw

>0

,

(23)

u,

.

,

u,

(v, v)

288

XII. SINGULAR A N D RECTANGULAR OPERATORS

w

S,

w (u - u,)

a

u, (u

-

uo - orw,u - uo - aw) = (u - uo ,u

-

uo) - or*(W,u - uo)

- 4 u - uo , w)

3 (u

S, ,

+ I a I2(w,w)

- 110 9 u - UO)

a,

(w,w) # 0. -Kw,u - uo)12 2 0 (w,w> (w,u - uo>= 0

u

- u,

w

S,

.

u

u u

+ (u - uo)

(u - uo) S , .

uoE S ,

Theorem.

= uo

If u

of

w u

of

T, of

ifu

T.

(u,W)

to

u

w,

v.

= (Tv, W) =

(v, T#w)

T#w

w

by

T#

T

u

Tv,

(v, T#w)= 0,

K T#w = 0,

w.

u of

(u,w)

=0

(28)

5.

T#w = 0. u

u

Ty,

-

u,

=

+

= u1

289

EXAMPLE

T

S,

112,

u1 E s, ,

u2 E

S,

s, T,

x

u2

y (U - 111,

Ty)

=

(T#(u - uI), y)

==

0

y T#(u - ~

u

-

u, u

u1

-

S,

u

-

ul

u

(29)

u

(u,u - Ul) = (u - u, , u - Ul)

ul

0

1 = )

+ (Ul , u - Ul)

u1 = u2

S,

(u -u] , u -ul)

=0

u

-

=0

. (30)

u

T

u,.

u

T, on u

by

T#

T#. u.

5. EXAMPLE

T n x n

0 1 0 ...

'

0 0 0 0 0 0

.*.

290

X I I . SINGULAR A N D RECTANGULAR OPERATORS

u=(

0 0 0

T~u =

.**

1 0

('i m

TtT v

=

diag(0, 1, I ,

Ttu,

...,

S, :

by on

6. CONCLUSIONS

T

u = Tv or

u

29 1

EXERCISES

by

Exercises

1.

0

-j

-j 0

-1 j

-1

-1

1

A=(-!

-J

( 1; ) 1

Ax

=

2j - 1

2.

a,

AX = U

(a)

a

= col(1,

-1, -j,j)

a = col(1, 1 , - j , j )

(c) a = col(1 + j , 1 - j , 1 - j , -1

-j)

(d) a.= col(1, 1 - j , -j, 1 - j ) 3. (Caution:

UV~,

?

292

XII. SINGULAR A N D RECTANGULAR OPERATORS

5,

4.

=

uvtK,

K

EX = y x

y

y

5.

Px ?

=a

?

u.

CHAPTER Xlll

The Commutator Operator dW - -- [S,w] = sw - ws dz

S z.

by dx _ - sx dz

x

K,

S W

x

(3)

= xytK

(2),

y dz

do

= @ytK

dz

=

+x

K

SxytK - xytKS

=

=

SxytK

+xyWK

[S, w]

(4)

W’s,

W

293

294

XIII. THE COMMUTATOR OPERATOR

W

.

(l), dM _ -- SM, dz

S

M(0) = I

(5)

z,

W,

W (6), by

W,

W. (5),

S

S

by dSdz

[A,Sl

(7)

A A

Lie book.

1.

295

LIE GROUPS

1. LIE GROUPS

3)

(A, B,...) 1.

I

2.

X

3.

X-l,

not distinct jinite dimensionality.

X

x = X(ff,, a l ,..., a,<

so

A

..., ffk)

(8)

so

a l ,..., ak ,

X X

by do

296

XIII. THE COMMUTATOR OPERATOR

global-i.e., local Lie

continuous n x n

As

of n x n of

n x nI

X

X-' GL(n)

general linear group

n x n

SL(n) unimodular group. n x n unitary group.

special linear group),

U(n), n x n

n x n

51 O(3)

O(n).

by

rejection-rotation

of

up

A do

representation

of

2.

297

INFINITESIMAL TRANSFORMATIONS OF A LIE GROUP

faithful

A

As by 1 .

AB =

=

C

by

1,

GL(n) n.

by realization

by

14. realization

2. I N F I N I T E S I M A L T R A N S F O R M A T I O N S O F A L I E G R O U P

X(0, 0,...,0) = I

(9)

..., ak ,

X

al,

X aX/aa,,

a1 =

= ak =

0.

298

XIII. THE COMMUTATOR OPERATOR

by

S

zo)

zo .

z

zo) zo ,

S. zo ,

z

=I

+ z a i X i + .*.

(12)

( z - zo).

a,

(1 1)

z

= z,,

,

ai

S S

S(z) a,.

S(z)

by

Xi. S a1 = a2 =

by on S ( z ) .

S(z)

= ak = 0.

3.

299

ROLE OF THE COMMUTATOR

3. ROLE O F T H E COMMUTATOR

M

N

by

aii

Pii

on

#Iij Qii N

M

N

M-lN-lMN.

300

XIII. THE COMMUTATOR OPERATOR

(17),

i

by

j.

do

of =I

M-lN-lMN

+ 2 y,Xi + *..

(19)

&

ai

of

[Xi , Xj] =

2 CfjX,

(20)

k

ctj

by

structure constants

cFj

4. LIE ALGEBRAS

algebra

A(BC) = (AB)C

associative algebra. linear b

(aA + bB)

5.

301

THE PRODUCT RELATION

B, by [aA

+ bB, C] = a[A, C] + b[B, C]

(21)

b

u

(22)

[A,Bl = -[B,Al

(B)

[A, [B, Cll

+ [B, [C, A11 + [C, [A, Bl1 = 0

(23)

(24)

[A, A] = 0

[A, [B, Cll - [[A, Bl, Cl

= [[C, A17

(25)

BI

5. T H E PRODUCT RELATION

;x (G x 5)+; x (5x"u) +; x 6 X Z ) ++ + =;(; * 5 )-;(; * G) + w(w * u ) - u(v -+-3

-3

w)

+ u(w -3-3

-3

-++

-

-3

w) - w(w u )

=0

302

THE COMMUTATOR OPERATOR

by by

[A, B] [A, B]

= AB

-

BA

AB [A, [B, Cll

+ [B, [C, A11 + [C, [A, Bl1 = A(BC - CB) - (BC - CB)A

+ B(CA AC) - (CA - AC)B + C(AB - BA) - (AB - BA)C = 0 -

(26)

n x n n x n

full linear algebra of order n. not

L(2).

K

2 ;( ’

bg

-

y

(a

cf

= (c(e - h ) - ( a

-

= AX

d)g

-

d)f - b(e - h) cf - bg

1

6.

K-SKEW-HERMITIAN

0

-c

c

0

o

c

b

-a+d -b

4 x 4

303

ALGEBRA

"i

-C

0

~5'~).

no n x n

L(n), n x n

a

n x n n x n

An--l,

go on on

by go

by embedded

6. K-SKEW-HERMITIAN ALGEBRA

by s = xtKx

(27)

304

XIII. THE COMMUTATOR OPERATOR

K

R

S

= $3

(27)

S do do

A

+ AtK = 0,

+ [A, BltK

K[A, Bl

KB

=

-

-AtKB

(28)

+ (AB BA)tK KBA + BtAtK - AtBtK + BtKA - BtKA + AtKB = 0

= K(AB - BA) = KAB

+ BtK = 0

-

[A, B]

S = -jR S

of

z,

K (27)

of

7. T H E Ad OPERATOR

L

A, by

(29)

7.

305

THE A d OPERATOR

, A, by AdAX

(30)

=

X adjoint operator.

adjoint,

11.

D

derivation.

{A, B} D{A, B)

= {DA, B)

Ad,(BC) = [A, BC]

+ {A, DB)

(31)

= ABC - BCA

=ABC-BAC+BAC-BCA = [A, BIC

+ B[A,

= (Ad,B)C

+ B(Ad,C)

(32)

inner derivation,

by

L

X AdAdAB =

[Ad, , Ad,]

L. = Ad,

Ad, - Ad, Ad,

Eq. AdAd~6X= [ A ~ A BXI, =

=

-[X, [A,B]]

[[A, Bl, XI =

[A, [B, XI]

+ [B; [x, A11

(33)

(21),

306

X I I I . THE COMMUTATOR OPERATOR

=

[A, [B,Xll - [B,[A, XI]

=

(Ad, Ad, - Ad, Ad,)X

8. LINEARITY

Ad,(aX)

=u

(34)

Ad,X

a

all

1 of

111.

Xi

hi Ad,&

=

[A, Xi]

(35)

= Xixi

n x n

4,

(Ad,)%

= Ad,(Ad,X)

(A~A)= ~ XAd,(Ad.~'X) (Ad,)nX = Ad,(Ad:-'X)

=

[A, [A, XI]

=

[A, [A, [A, XIII

(36)

9.

307

EIGENVALUES A N D EIGENVECTORS

(Ad,)OX

=X

A

X.

:

of

semisimple

Xi n x n

A AdAXi Ad,&

Xi,Xi,,Xi,,...

= hiXi

+ xi = &xiz + xil

(37)

= hixi1

AdA&,

hi m i ,

hi mi.

body do

on An-’

of

on 9. EIGENVALUES A N D EIGENVECTORS

of

A

A?

308

XIII. THE COMMUTATOR OPERATOR

A, ui

=

.

(Xi - Aj).

Wij W,

by

at least,

Wit. A

ui

hi : Aui

= X,U~

vi

111,

V.tU.

= 8.. 13

vitA

= hivit

a

3

= uivjt

= Auivjt - uivitA = (hi- hj)uivit = (Xi -

A

Wij A

.

16,

9.

EIGENVALUES A N D EIGENVECTORS

Au,

= Au,

Au,

= Au,

309

+ til

111,

17,

2, VltU, = v,tu, = VltUZ = v*tul

x, =

(WI1 + WZz)

=0

w21

x, = Ad,& XI

1

=

= AdAX, =

w1,- wzz

(40)

-2W1,

3

.

310

XIII. THE COMMUTATOR OPERATOR

10. T H E EQUATION U

= Ad,V

u = Ad,V V,

(41)

U. Y

(7),

A A. A

V,

A,

of

A

no

2. no of A.

0

X

(36) by B(Ad,)U

= o(Ad,)

AdA2X

= Ad,X =

V

v ,

A, A

4 of

10. THE

31 1

U = Ad,V

11,

3.

111,

A

B (A, B)

= tr(AtB)

(43)

A.

=

C (A,zBji)

= (B,

A)*

(45)

X

(x,Ad,Y) (X, AdAY) = =

=

(Ad,#X, Y)

Y (47)

Xt(AY - YA) XtAY -

XtYA

(48)

312

XIII. THE COMMUTATOR OPERATOR

of

by

not,

=

=

=

(50)

Eq. (48) =

-

=

-

=

-

(51)

=

A

Eq.

, we

V.

Eq. (41) 11. T H E KILLING FORM

on form

Killing

scalar product of Cartan.

of no

(53)

=

by

of

Eq. go

B) = k

(54)

12.

T H E EXPONENTIAL

313

Ads

on (54)

11. go

12. T H E E X P O N E N T I A L O F AdS

dW - = AdsW dz

S

=

(55)

So, W

=

(56)

AdsJW,

on

z

S, by LSX RsX

=

sx

=XS

on n x n

314

XIII. THE COMMUTATOR OPERATOR

=

=

(60)

11, -

=

(61)

(62)

=

=

-zS0)

=

(63)

S = So,

= =

(64)

=

13. SIMPLE NONUNIFORMITY

(2),

by

simple

S

nonuniformity by

A. A can A

S

dS/dz

A.

A, by

s=

(65)

So = S(O), -=

dz

-

13.

315

SIMPLE NONUNIFORMITY

w = ezAdAX Eq. ezAdA

dX AdAX + ezAdA__ dz

= (ezAdAS,)(ezA3AX)- (ezAdAX)(eZAdAs,) = ezAdA(Ads,X)

by Eq. S

X =

(69)

Ad(s,-A,)X,

Eq. W = =

Ada)

Ad,so-A))&

A~A)

Ad(s,-~))Wo

=

-

-

Eq. x =

-

Eq. (5) -

=

go

Eq.

Eq.

S A

Eq.

dA -

dz

B

(72)

-

(73)

on.

316

THE COMMUTATOR OPERATOR

S S

on

14. T H E E X P O N E N T I A L L Y TAPERED TRANSMISSION L I N E

by

-JP.

S

Z d Sdz - -jPK(-l,z

K

=

0

z/z A.

zc - (BIZ) = 0 A-D=K

A

=

=

0

K

15.

317

K

Z

= eKZZO

x = 0.

2,

K

zA

z ( S o - A) by

1 K yz - -

ez(S,-A)

2Y

=

-j B - Z, Y

yz

yz

?/2 + 2 - I

2 4K

K

=

0,

15. CONCLUSIONS

on

by on

,

,

LA.

+ -2 Y1 K

yz yz

318

XIII. THE COMMUTATOR OPERATOR

by simple nonby

uniformity.

Exercises

1.

Xi,

pi

Ad&

A

(a)

=

0 1 ( I o)

A

= pixi

(c)

=

2.

A

=

[ 8,

~(x),

by (AdA)X

X.

=0

~(x)

U. 3. A=

c a 0 2 0

.

4.

by

M=(

0

case

319

EXERCISES

5. 6.

i] ( ; ) [ P A ,Dn-*B]

Dn[A,B ] =

k=O

eD[A,B ]

=

[eDA,eDB]

eD on

(64) Leibniz' rule.) 7.

s= a, b, 8,

(

0

j b cos 8

j b cos 8 b sin 8 0 0

u cos 8

-jw

0 -ju sin 0 -jusine 0 -b sin 8 0

z.

w

1

X, , X, , X, S = X,

cos B

+ X,sin 8 + WX,

X, , X, , X, by x l f =1 ~ c 0( -1o), l

x i = -1 c (O 2

j

),

j 0

X 3 ' = -1 ( i 2 0

S'

c

K' S

X,',X,',X,' dW - -- [S,W dz

by X, , X, , X, ,

Wo

-dW' - [S',W ] dz

0) -j

S on

K

320

XIII. THE COMMUTATOR OPERATOR

8. Solve for the matricant by the methods of this chapter, when

/3 and y being real constants.

CHAPTER X I V

The Direct Product and Kronecker Sum direct product Kronecker sum

book. do 1. T H E DIRECT P R O D U C T

on by

u

y,

x

'1 '1 by

u by k 5

y,

5

k,

v by 1,

x by k

by k. 32 1

y by 1,

3 22

T H E DIRECT PRODUCT AND KRONECKER SUM

4x 4 Eqs. XlYl

+ B,u,)(A2v1 + B2v2) = A,A,(u,v,) + A,B2(u,v,) + B,A,(u,v,) + B1B2u2v2 = (A1u1

A, Eq. product

A,

(4)

(6)

by A, . direct product Kronecker

x A,

(7)

A, A, A

p

= (aij),

B

= (bij)

q

(tj)

by

6.. 13 = x . y3. E

3

Tii

upon = UiVi

(10)

2.

JUSTIFICATION OF “PRODUCT”

323

2. JUSTIFICATION O F 44PRODUCT*’

A.

(A+B) X C = A X C + B XC A x (B + C )

=A

A x (B X C)

x B +A

= (A X

XC

B) X C

(13) (14)

(15)

C.

o=oxo I=IxI

(A X B)(C X D)

{(A x B)(C x D))ii.mn =

= (AC) X

(BD)

2~ x B ) ~ ~ , ~ x~ HD)rs.mn) (c TS

= {(AC)irnH(BD)+n}

= {(AC) x

AB=C,

(BD)lij,mn

MN=P

(16)

324

XIV. T H E DIRECT PRODUCT AND KRONECKER SUM

C

N,

n x n

(A X M)(B X N)

= (AB) X

m x m,

(MN) = C X P

(18)

=

x

Eq. /I,

01,

(B x N),

x

y

N

=

=

B x

N,

C

a/? = y

B

Eq.

(C x Eq.

(A X M)(B X N)(C X P)... = (ABC...) X (MNP...)

B,C,...

N,

n x n

...

(19)

m x m.

3. T H E PRODUCT OF MATRICANTS A N D THE KRONECKER SUM

sum,

up

? by dM dz

-

dN -dz

-jm,

M(z,)

= I,

-jBN,

N(z,)

= I,,

xN d

dz (M

x N) = =

dM dN (x) x N + M x (x) -j(AM) X N

- jM X

(BN)

(21)

4.

d

- (M X

dz

N)

=

-j(AM) X (1,N) - j(1,M) X (BN)

=

-j(A X I,)(M X N) -j(I,

=

-j(A X I,

Eq.

+ I,

X B)(M X N)

=A X

=

(22)

I, +I, X B

z1= z 2 ,

N

X B)(M X N)

x N) C: C

x

325

GROUP THEORETIC SIGNIFICANCE

(23)

(M x N)

z,

Im+, Eq. (22)

Eq. (23).

B. N, Kronecker sum

Eq. (23)

B.

4. G R O U P THEORETIC SIGNIFICANCE

by

by

326

XIV. T H E DIRECT PRODUCT AND KRONECKER S U M

do by

A n x n S by S' =

=

n 0,),

n x n A

by

S by

S'

N

N

(20).

by

zo ,

x

S",

by

zo ,

0,

z

M N

= I,

= I,

+ €A + EB B

A

E

(14),

(MX N) = (I, = I, -

+ €A) X (I, + EB) X I, + E{A X I, + I,

I,+,

+ €{A X + In

Im

X

X

B} + E'A X B

B}

(25)

E.

zo ,

x N) by

5. EXPONENTIATION

by

N.

5.

327

exponentiating

ec = eA

x

x

= (AI,)

=

=

=

(27)

x

x on

C,

C

+C +

ec =

1 2 1 ~ 2

x eB = =

+

Bo =

Ao = eA

eB

1 + ++ .-Ix 1 - + + 21! + ..-I 2!

1,

~2

22

x

r=O s=O

Y

=

k

=p -

k

p

s

k.

k

B p,

328

XIV. T H E DIRECT PRODUCT A N D KRONECKER

6. EIGENVECTORS A N D EIGENVALUES O F THE DIRECT PRODUCT

by

A n

m,

(;i) Eq. (32) (A x B)u(”’) = ( X r p s ) ~ ( r * s )

(34)

A x

t1(+l8)

A d r ) m yes), A x A x of

A

A

Arp8. n

(mn)

A x

~

(

~

A

1

~

)

8.

329

NECESSARY CONDITION

A x

A x

A

7. EIGENVECTORS A N D EIGENVALUES O F T H E KRONECKER SUM

(23)

Cij.kh

+

= Aik6jh

6ikBjh

aij

i

=j ,

B

A

A hh

C ~ ( 7 . s=) (A X 1 + 1 X B)u(T,s)= (4 ~

(

~

9

~

+ p8)u(r*s) of

)

A

of

A A A

8. NECESSARY C O N D I T I O N

C (24), P,

A A

C

(35)

(24)

3 30

i

=

pk

XIV. THE DIRECT PRODUCT AND KRONECKER SUM

1, ..., mn. p i ,i = 1,

qi ,

hi , i = 1, ..., m,

..., n,

Aj

qi

*

q, , q z , q 3 , q 4 .

4x 4 A,

A,

pz

p,

4 + Pl = 711 A, 4-k = 7 2 4 + P2 = 713 A2

+

P2

= 714

on

on

71

q i do

- 712 - 713

+

73 =

0

no

A

B. q i do

no

on of

z,

33 1

9.

z.

by book. 9. MIXED INVERSES

As

X AX+XBt=C

(36)

on

A

Bt, by

C A,

Bt

=

0,

A = 0. A B

B

Ct

(36) (36)

on B

B

X

(36)

by

z(Ax1

Bt.

(37)

+ 1 x B*)ij,khXkh

= cij

kh

X,,

n x n

Cii no

(A x I

n2 x n2

(Xi

+ pi*)

(A x I Xi

+ I x B*)

A

pi

+ I x B*) B.

332

h,

AND

+ ps* = 0.

C,

Eq. (36)

B

no

A. Eq. (36)

by

(AX + XBt). AX = AX BY =PLY xyt A(xyt)

A

+ (XYW(A + p*)xyt =

Bt,

10. SUMMARY

book,

As

Eq. (36).

(AX + XBt),

Exercises

1.

A

n x n (A x B)-l = A-’ X B-’

(b) x B) = B) (c) A X B = I ImJB In

B

m x m

333

EXERCISES

2. (A X I

3.

-

I x A). (A x

A

4. 1 0

1 -1

(A x

c=(o 1)’ (C x

5.

D=(-2

1)

(16)

0 1 A=(l

o),

O ab) B = (a/b 0

A (A x

x

A ?

‘1

6.

6 4 3 2

A = ( 83 2 6 4

4 1 8 2

7.

b)

1 0 1 0

8;

A=(; 0 1 0 0

2 x 2 ?

8.

AX

X,

+ XBt = C

334 9.

XIV. T H E DIRECT PRODUCT A N D KRONECKER S U M

tl

v

by u

uluz , uZ2) u12u2, u1u2, uZ3)

=

AV v1v2 , uZ2)?

A l zl,

,

v12v2, v1vz2, vZ3)?

AL2I = A x A, n x n

CHAPTER X V

Periodic Systems

on

1. REDUCIBILITY IN T H E SENSE O F L Y A P U N O V

n x n

z.

dx - dz

X(0) # 0

-jRX,

X

n

R(z)

X(0) =

X(z) 335

(2)

336

PERIODIC SYSTEMS

by x

= L(z)y

(3)

x = L(z)Y

(4)

n x n

Lyupunov

bound.

reducible in

sense

ofLyupunov

A (4)

y

= e-iAz

X

= L(z)e-iAz

good

(7)

do

2. PERIODIC SYSTEMS

R(z)

I, R(z z,

+ 2 ) = R(z)

x

(9)

2. (2),

z by ( z

+ I):

-jR(z)X(z

=

+ 1)

X(z + I )

C

337

PERIODIC SYSTEMS

(10)

1

Eq.

X(z): X(z

+ 1 ) = X(z)C

C

(11)

A

c = e-iAl

(12)

so c z l l

= e-iAz

C(z+Z)ll= e-jA(z+l)

~

e-jAze-jAl

- e-jAzc

by

CzIi

(13)

C

X(z).

L(z) = X ( Z ) C - ~= / ~X(z)eiAz L(z

(14)

+ 1) = X(z + l)ejA(z+fl) = X(z)CC-lej*Z =

z.

(15)

L(z)

L(z),so

X(z)

= L(z)e-iAz

(16)

R(z),

A

338 so

XV. PERIODIC SYSTEMS

L

* dz

= j(LA - RL)

(5).

A,

(18)

L so U

u = LV L (2)

(19)

X by X,

U

dV - -L-l- dL L-qJ dz dz =

(-L-l

=

-jAV

+ L-'

(14). dU

dz

dz dL -jL-'RL)V (20)

X,

L,

-jAz)

A 3. F O R M O F T H E F L O Q U E T FACTORS

on

K

R(z)

K-

+ I) (1

C

= M(z)-'M(z

+I)

(21)

5.

339

THE FLOQUET MODES

N

M (M-'N)tK(M-'N)

= NtMt-'KM-'N

= NtKN = K

C

A,

K

= Mt-lKM-l = Lt-l

(12),

exp[-jAtz]K exp[jAz]L-l

= Lt-lKL-1

(22)

L-l,

R

4. DETERMINATION FROM THE MATRICANT

z

=

0, L(0) = I

(23)

M(1) = L(l)e-jAl = e-jA1

(24)

A,

-jAz).

no

5. THE FLOQUET MODES

of A,

of

A u ~= Piui

X(Z'(0)

= ui

x ( ~ ) ( z= ) M(z)ui = L(z)e-jAzui = e-'flvz{L(z)ui}

(25).

(26)

340

XV. PERIODIC SYSTEMS

Floquet mode ui 1.

L(z) Lui ,

Eq.

x(s)(nl) = e-%inlU,

(27)

6. SPACE HARMONICS

L(z)

1,

z

L,

=

I1

1

L(z)eiznnz'l dz

0

Eq. x(i)(z) = e-j~,z

2

~ ~ ~ - j z n n z / l U ~

n

up),

harmonics,

@'.

K, A

/Ii by

space

A. on

pi

7.

34 1

ORTHOGONALITY RELATIONS

pi 2r/l

(31).

uy)

do,

pjn).

7. O R T H O G O N A L I T Y RELATIONS

R,

A, ui

u ~ ~ K u= , ~oiSij,

ui

= *1

(33)

(26), ~ ( ~ ) t K x ( -= j )expb/3,*z] exp[ -~/3-~z]u~tLtKLu,~ = expL(fii* - /3,j)z]uitKu,j = eXp[j(B-i

-

- 0.8.. I

L

(34)

13

pi* = {x(i)}

x.

K (34),

(32)

2 e ~ p ~ / 3 ~ ( ~ ) z ] uexp[ : ~ )-$FJuK) tK n,m

=

2expU(/3,"i)

-

/3q)z]~:~)tKuL?)

n,m

= 0 1. 8$3. .

i

(35)

=j

ujn)

ui

~UY)~KU =F ui ~ n

(37)

342

XV. PERIODIC SYSTEMS

A, A do

,81n) R(z)

Pi.

by

=

1

1, 1

R,

=

-j

2

dz

8.

343

EXAMPLE

n,

by

(31),

pim) + 2mll = &m+n)

=

(41)

CRnpnLuim) m

or

(42) !(Rnpm - rB(im’8n,mI)!= 0

(43)

n

m

N x N on

R(z) N x N

(43)

N x N (43) (43) by

(42) by

8. EXAMPLE

0

R=(

(44)

K=I fl

y

13, by

8.)

344

XV. PERIODIC

R

=

R(z) = Rle-jSz

+ R-lej@z

+

=

27r/t3,

(42) R1u;-' - /9;~3 R_,U;+'

i

0

pi

n.

(45),

u;O) # 0. = pl"'p;-"

Sl"' of

=

by

&p

up). by R,

by

y2

(45)

=

p(p- 8)

& l/pS.+4y2) /3,

R-,

8.

8.

8(8,

-

345

EXAMPLE

8,

y2

P-)

8. n-mode operation.)

8,

Eq.

u:’

k+ n = 1,

Eq.

ui1)

Eq.

c1

n

Eq.

= 2,

Eq. c2 =

c1 =

0.

0.

8-

Eq. (32), x + ( 4 = k+

i

(B+ + B)

346

PERIODIC

R

Eq. ( I ) (49).

/3-

/3+

(44),

9. CONCLUSIONS

Eq. (44) do

R

Exercises

1.

L

3,

As

LtKL

=H

H

R

2.

A

3.

on

R

=

-.a)

?

4.

R,

by

/3

K

=

(dZ/dz)/Z

347

EXERCISES

?

by

_ -5.

R = (a a, 6,

/3

Eq. (47).

-

0 2b

PIo).

=

a

+ 2b 0

Bz

up

/?z

1

CHAPTER X V I

Application to Electromagnetic Theory

by of

A

do

x,

1. CARTESIAN SYSTEM

E E = ( 2 )

H,D,

J.

A,

do A

=

(-a/ayalaz

o

alax

348

-apx o

Q-

1.

349

CARTESIAN

by

E.

d, V,

E.

apz)

6’ = ( a p x

(3)

E,S’E,

6‘

T. ajax =

k;2)

(4)

(ST)

6. 6. (ST), T.

cpST

A Cijk

eijk =

0

,

ijk 123,

-

1

ijk

odd

on

AS A6 6’A

S’A

=0

(5)

0

(6)

E

of

by A’

= 66‘

-

6‘61

(7)

350

APPLICATION TO ELECTROMAGNETIC THEORY

x

x)

=

-

V2.

16’61.

2. MAXWELL’S E Q U A T I O N S

AH

==

+ -D at

J

(8)

d E = --B

at

6‘D = p S’B = 0

D

+ P = EE

= eOE

B =poH

+M

= pH

no E

p

J

E

p

p

AH

=j

AE

==

w~E

-j~pH

S‘E = 0 6‘H = 0

(14)

3.

MAGNETIC HERTZIAN VECTOR POTENTIAL

35 1

p

E

by

d

5)

on A2E = =

-

(14):

6’6)E

-jwpoAH

= w2p0q,E

(18)

+w

~ ~ ~=E0 ~ ) E

(19)

+w

~ ~ ~= E 0~ ) H

(20)

3. MAGNETIC HERTZIAN VECTOR POTENTIAL

n,,, E = -j w An,

(21)

by H

= W2pEnh

+

S$hh

(22)

J *!,I

5) 6(6’nh

- $hh)

=

+

W2pE)nh

$hh

+ W 2 p € ) n h= 0

(24)

(22), H

=

-6’6nh+ 66’nh= A2nh

(25)

352

TO

(6),

by

(23). (25)

by

nh, (24).

4. ELECTRIC HERTZIAN VECTOR POTENTIAL

He.

H E

=jwc

All,

= w2qJIe

-

4,)=

*,

=

(26)

+ + w%p)IIe

(28)

6'n,

(29)

+ w%p)II, = 0 (29)

(30), E

=

+ 66 ll, = A2'II,

As

p

(31)

(30).

IIe

E

(30)

5.

353

CHANGE OF BASIS

E,

p,

u.

3)

(8)

p

by

do no 5. CHANGE O F BASIS

0 = tan-’(y/x) Y

=

(2+ y

y

z = z

We

.. (32) =

SE S

4)

6

(1 5)

E=SE

R=SH d

(33)

2

2 = sos-’

(34)

354

XVI. APPLICATION TO ELECTROMAGNETIC THEORY

(14).

on

6'

by

6

A2

=

8 = S6

(36)

8s'

(37)

-

Sd' 8S-l S

=

I.

6'6 S

S

S-'

S

8' 8. S8'8S-1

by

S8'8S-l.

6'6

(Ss'dS-1

(30)

+

W"€)rIr,

II, .

=0

(24),

(38)

6.

355

POLARIZATION COORDINATES

6. POLARIZATION COORDINATES

of

polarization

by z

E,

E, .

Eq. (39), - = - - ax ar, a ~ ax ,

++-dr,av

a ay

356

XVI. APPLICATION T O ELECTROMAGNETIC THEORY

Eqs. (34)-(36)

i($) a

(45)

S

(6). Eq.

(20), (24),

(30).

Eq. (39), r

eje = cos

0

= d x 2 + y2 =

+j

0

= (x

d

E

+ jy)/r =

(47)

(48)

Fn(r, 8, 2)

7.

357

CONCLUSIONS

2,

(46), on F, S’SF,

= -(y2

+ Bz)FrL

(51)

7. CONCLUSIONS

(2), do. on by do on

358

XVI. APPLICATION T O ELECTROMAGNETIC THEORY

do,

Exercises

1.

Eq. by

2.

x =Y

0

p

y

0

p

=Y =

S

cos

e

Eq. (33)

s= 3.

A , 8,

4.

A , 8,

i

0

by 0

cp

p

cos

0

cp

p

8' 8'

5.

v = pv - ZI + t - (Vp) + v x (pt) = x t )+ x + v (t x ); = ;- (V x +u ) - +u * x v) (I$)

6.

(z .V)

(;V

V ( t * G)

v x (t x ); 7.

do

a ) ?

= =

tx

-

Eqs. (5)-(7) by

x );

+ (Z

-$

*

-

+ v' x (V x t ) + ;( t )+ - ); *

by

*

V)t

CHAPTER X V l l

Sturm-Liouuille Systems d ( p -) dz

p , q,

- qf - hYf = 0

dz

r

(a, b),

z,

f(z) z =a

z = b.

z z

p

of q = 0.

Y

of

As

N (a,b) z, = a

+ nh (n = 0,1, ..., N ) ,

a = z,, , b = zx , h = ( b -

by z,,

, h

by

(N + 359

a)/N.

360

XVII. STURM-LIOUVILLE SYSTEMS

-+

00.

on bound on

N on

x,

+ 00,

1. A P P R O X I M A T I O N IN A FINITELY D I M E N S I O N E D VECTOR SPACE

x,

on

n = 1, 2,

...,

n

=

by z.

x.

X

p, q,

Y

upon x

f(z).

by

f(x) by

0

1.

36 1

FINITELY DIMENSIONED VECTOR SPACE

do on

x,

.

n

zn

==

2 dn

(4)

1

d,

.

d,

+

= f(zn)

t dn+lf’(Zn>

f(zn+l)

= f(zn

f(zn-1)

= f(zn - dn) = f ( z n ) - &f’(zn>

dn+l)

f(zn) =

(6)

(7)

f n

n n

d,+,

(5)

=

1

=

fo

N,

fN+l d,

= x1

-

xo

by

f , = f N = 0,

f,’

1

=f2/d,

fl’ = d,u - 2 -fd

f N ’ = -fN-,/dN.

362

XVII. STURM-LIOUVILLE SYSTEMS

where dN+lis taken as appropriate to the boundary condition. We write T

>

fl f 2

fN-1

_- 1 dN+l

f N

1

I

which can be written as

f+' = D+f where

1

=( F ) ( - a i i 7t1

Similarly, from Eq.

+

8i.j-l)

we have 1

fl' = -dl f1

f;

1

=

-(f* -fJ 4 1

fN' =

which we write as -

1 -

d,( f N - f N - l )

0

0

fl f 2

f,

fN

I

\ I

(14)

363

2.

DD+PDf - Qf

-

hRf

D-PD+f - Qf

- hRf = 0

=0

by R-l, Hf

(18)

= hf

H

= R-'(D+PD- - Q)

H

= R-'(D-PD+ - Q)

D-

H. 2. MODIFIED STURM-LIOUVILLE E Q U A T I O N

by g

= p1My1I4f

(21)

Y

w. w

V(w)

w

364

XVII. STURM-LIOUVILLE SYSTEMS

D+D-

d2/dw2

D-D, w, (w7‘ =

1 + h) = gn + hgn’ + 2! h2&’ +

gn+1

=

gn-l

= g(w, - h) = gn - hg,’

(

-

D,

=

h2 I

+ --2!1

h2,”

by 2 1 1 - 2

0 1

-!

A -;

0 0

-2

. Hg

H = D,

= Ag

+ diag(Vl , V, , ...) v,-2 1 0

1 v,-2 1

0 1

v3-2

+

)

(26)

3.

365

THE CHARACTERISTIC EQUATION OF H

no (24).

by 3. T H E CHARACTERISTIC EQUATION O F

H

by

lv1i2-h v z - 2 - x

0 1

I

Vn-2-h

(29)

n x n.

k x k

n -K

by on

V1-2-h

1 pk(h)

1

v2- 2 - h

=(vk-2 vt-,

V1-2-X

-2

1 v*-2-x

!

= ( vk

I

I

-2 -

-pk-2 p,

=

v, - 2 - x

p 2 = (V, - 2

- h)(VZ- 2

-

-

1

V , 's

-h

366

XVII. STURM-LIOUVILLE SYSTEMS

(30)

Jucobi matrix V k

(30) do

V, do by 4. STURM CHAINS V k

p,(h)

p,(X) n,

chain h

pk(h)

=

0,

(a

pk+l(h)

< h < b)

k,

pk-l(h)

k,

po(X)

p,(X),

no

p,;(h)

(30)

by

by

Po(4

k pk(h)

k.

2

pk+l(h)

(30),

h,

p,(h) (30),

A.

p’s Pk

=

=1

pk-l(h)

=

0

h,

A.

pk-2

R(x) (a,b), I s R(h), - co

h

0

n(u) n.

R(x)

+ co,

+ co a

b, pk(a)

k

- 00,

4.

367

STURM CHAINS

pk(h)

pn-Jpn p, pi

b

piPl i # n,

n(b),

i

pnPl(b) > 0,

=

p,(b)

n,

+

0

-

b

by

n(b)

by

p71-l

by

p, , pk(h) pk(A) n(a) = 0. As h +

n(b)

+

up

(30),

by

h -+ - 00,

$),(A)

00,

k

b

odd.

P,~(A)

pn-l/pn P,-~

n. -n.

p,

on

k

n n

k

0 odd,

pk ,

pk-1 .

pk pkPl

A,

$1 n(a)

A

n(b)

V(x) n

(n -

n. V(x)

368

XVII. STURM-LIOUVILLE SYSTEMS

5. R E C U R S I O N FORMULA

(30),

on good

xn = anxn-i

+

n.

b,

a,

(33)

bnxn-z

do do Yn

(34)

= Xn-1

(33) x n = anxn-1

(34)

+

(35)

bnYn-1

(35)

P, = P,

P by

(n -

P,

(36) n. a,

n.

b,

(36) x,

n

.

yn by

odd t i n = Pnpn-1un-2

(38)

5.

369

RECURSION FORMULA

P,P,-, P,

a,,

b,

. (36),

by x,

on x,

x,-,

y , by on n.

1).

XV,

(34),

As Yn = knxn-1

k,

(36)

bn = kanan-,

k

k,

b, ,

a,

.

a,

(33), x, = anxnPl

nna,

=

+ kanan-,xn-,

P,

x,

(44) W,

= wn-1

+ kwn-2 P,

(43).

3 70

XVII.

do

do

Exercises

n x n

n

1.

Jn =

a, b, Jn

c

. (Hint:

3

J, .

(Hint:

a

( I 1)

dt

2.

d,

D, ?

R,

3. R4

+ aR3 + bR2 + CR+ dI = 0 Rn.

(Hint:

R” = fnR3 + gnR2+ h,R

f, , g, , h, , k,

+ k,I

n a =

b

= c =

0, d

=

-1.

CHAPTER X V l l l

Markoff Matrices and Probability Theory

body do up, by

Markofprocess do

by

by

by

N(t). 37 1

372

XVIII. MARKOFF MATRICES AND ‘PROBABILITY THEORY

+ $No

t

=

$N(t)

+ 1).

t,

0, go

Problem of the Gambler’s Ruin.

?

1.

Markoff

random walk

373

STATE VECTOR

drunkard’s walk

1. S T A T E V E C T O R

up

(M

$0

+1

$M

$1

$M, $(M

on,

-

$0. no

on

no

pure state. do

$N

p N,

mixed state.

374

x

probability vector.

on

61.

11 ax 11

=

0111

x 11.

go by

!I x t- y II

= I/ x II

+ II Y II

(3)

2. TRANSITION O R MARKOFF M A T R I X

$n.

$m

pmn

pm,

up

.

m n.

P,~, #m,

by #n

2.

375

TRANSITION OR MARKOFF MATRIX

P,,~,

transition probability

n

m. $n.

do x, $n

pmnx, .

$m ym

$m

(4)

by m by y

= Px

transition

(5)

M a r k 0 8 matrix. s

by x(s).

xk x(s

xk ,

x(s

+

+

P.

There is some confusion in the literature as to just what is called a Markoff matrix. Sometimes pmnis taken as the probability that state m arose from state n, and the Markoff matrix is the matrix of these probability terms. In this case, each row must be a probability vector, since state m must have arisen from some state.

316

XVIII. MARKOFF MATRICES A N D PROBABILITY THEORY

3. EIGENVECTORS O F A MARKOFF MATRIX

xi x(s) x(s)

Eq. ( 5 )

= xi

PXi = xixi x(s

x(s

+ x(s

x(s)

+ 1)

(7)

+ 1) = xixi hi

P no

$M

..., 0)

(8)

xz = col(O,O, ...)0, 1 )

(9)

x1

= col(l,O, 0,

on by

x3,..., x,

up

on ~ ( 0= )

2 i

U a. Xa. - alX1

+ uzx, +

***

(10)

3.

377

m x(m)

=

Cui~imxi

11 x(m) 11 11 xiII

=

1

m.

Xi # 1

=0

(13)

hi 1 hi I > 1,

xi

A’s

m

of x(m)

x(m)

P

I hi I

<1 x1

x 3 , ...,x,,

m+m

(14)

for all X i

x2 Eq.

x(m)

x ( m ) = ulxl

+ u2x2 $M

u2 ,

on

a,,

of

378

XVIII. MARKOFF MATRICES A N D PROBABILITY THEORY

x1

x2,

by

(xl- x2),

(2x1- x2)

by by x1 =

(4 x2 .

I A2 I

x1 =

x2

1,

4. R E C I P R O C A L E I G E N V E C T O R S

yi yi-fxj= Sii

(19)

5.

379

NONNEGATIVE A N D POSITIVE MATRICES

y It p x3. -- A.3Yi tx.3 -- h.6.. = xi 3 1 3 y,tP

= piyit

Ptyi

= pi*yi

yi y.tpx. a E z p EYE . .tx. 1 = xiy,txi pi = hi .

P

P of

Pt.

5. N O N N E G A T I V E A N D POSITIVE MATRICES

A 30.

of

do

reducible

reducible reducible

A

C

irreducible.

380 by

T do

by by Theorem. A n irreducible nonnegative matrix always has a positive real eigenvalue A that has multiplicity one. The magnitudes of all other eigenvalues do not exceed A. T o A there corresponds a reciprocal eigenvector that can be normalized to have coordinates that are all real and positive. Furthermore, if the matrix has k eigenvalues, A, , A, , ..., Akp1 , all of magnitude A, then these eigenvalues are all distinct and are roots of

Likewise, all other eigenvalues are simple and occur in sets that are similarly related. I f k > 1, then the matrix can, by a permutation, be put into the f o r m

where the 0’s along the main diagonal are scalars or square matrices.

on no

$0

x

38 1

EXERCISES

1 x 1

(23) on

(n n x 1

x (n -

A A,

by A,

do do

6. CONCLUSIONS

on

do do on

Exercises

$200 $400.

1.

on

p $200 on

$400 ?

$100

(p >

?

$50 ?

?

( p < 8) ?

382

XVIII. MARKOFF MATRICES A N D PROBABILITY THEORY

2.

l b O O O

f !H

A=

'be..

l)

O c a b O O

O

c

a

O

a, b, c

a+b+c=l

P

3.

Q X = a P + ( l -u)Q

0

< < 1.

Q

4.

PQ.

u

5.

uvT, v A

(1 x1 11

Al =

A,, A,, 1, 11 x2 11 = (1 xg I(

=

1

x1 , x2 , ..., x, ,

..., A,, =

.*.= 11 x, 11

=

0,

A

CHAPTER XIX

Stabilitv

by

of

t ?

x(t) = 0,

stable.

x(t) unstable.

x(t)

x(t)

by

admissable. of by

do no

by by

of

x(0). z z

z

383

384

STABILITY

P P

P x ( t ) = exp(Pt)x(O)

(2)

100 do

1. THE BASIC THEOREM FOR STABILITY

Theorem. A system described by Eq. with P constant is stable for all initial conditions if and only if all the eigenvalues of P have negative real parts-i.e., are in the left-hand plane of A.

r(t>= s w S

(3)

(1)

_ du - (S-1PS)u = P’u dt

P P‘

by

(4)

S

P u(t)

M’(t), go

= diag(eAlt,eAzt, ...)u(0) = M(t)u(O)

(5)

1.

THE BASIC THEOREM FOR STABILITY

(4) A 1 0 0

O h 1 0

PI=(.

...

O

385

)

p!. = za pr!, z.+ l = 1

p! . = 0 t.3

(4) U(t) = M(t)u(O)

1

t

0 1 0 0

=o

t2/2! t3/3! t t2/2! 1 t

*")

(7)

j < i

(7')

j
=O

(8)

by

(PM),, .

(5). n t

h

tneat

386

XIX. STABILITY

Definition. P is said to be stable, or a stability matrix if all of its eigenvalues have negative real parts.

2. ROUTH-HURWITZ METHOD

P, P n

h

n

ak = 0 6,

a,

=0

k > 8.

k > *(n

- 1)

b,

p(h)

no Hurwitz matrix

H=

n x n

2.

387

ROUTH-HURWITZ METHOD

by

a,/b,

(bo/co) Routh matrix R :

R

=

b, b2 ...

b, 0

i

c0

C,

0 0 do

...

Theorem. The number of roots of Eq. in the right-hand plane-i.e., with positive real part-is equal to the number of changes of sign in the sequence a, bo do 9

9

9

9

---

Theorem. A polynomial, Eq. (9) has all of its eigenvalues in the lefthand plane-i.e., with negative real part-if all of the terms a , , b, , c, , ... are nonzero and have the same sign.

b, , c, , d o , by regular

by

E

no (A2

+h +

+ h - 1) = A4 + u3 + 2 0

0

0 1

1 - 1

H=(i

A2 -

1

=0

388

XIX. STABILITY

1, 2,

1, 2,

-1

of of A4

+

a 3

- h2 -

u- 3 =

(A2

+ h - 3)(h2 + h + 1) = 0

2-2

co 2-2

0

0

0

2-2

0

0

0 a-3

2 -2

(+

=

0

0 -2

O

O

+ - - -).

+ 6/a -3

B

B,

0.

co = E 2 -2

0

1, 2, c

0

0

0 0

+ b / B , -3 (+ + + + -).

-2

0

0 -2 0 0

+ 6/a -3

4.

CRITERION OF LIENARD A N D CHIPART

389

on do by

3. H U R W I T Z D E T E R M I N A N T S

dk,

k

n): A,

= b,

A,

=

Ai>O,

bo bl b, a, U , a2 10 b, b,l

i
4. C R I T E R I O N O F L I E N A R D A N D C H I P A R T

h3

+ h2 + h + 6 = + 2)(Xz - h + 3) = 0 =

-2

and &l j-jfi)

(14)

390

XIX. STABILITY

by

Theorem. All of the roots of Eq. (1 5 ) have negative real parts i f and only i f one of the following four conditions is true (if one is true all are true):

I-

%-2k+?

I1.

%-2k+2

111. a, a,

> 0, > 0,

> 0, > 0,

a,-,,-, an--2k-l

> 0, > 0,

A,,-, > 0, A,, > 0, A 2 k - l > 0, A,, > 0,

no

on 5. LYAPUNOV’S SECOND METHOD

by

< k < n/2 1 < k < n/2 1 < k < n/2 1
(17)

5.

39 1

Lyapunov function s

by (18)

s = xtLx

x

x

=

0,

s

t

ds -

dx dxt Lx + XtL dt dt = xt(PtL + LP)x = X ~ M X

dt

M Mt

= (LP

= LP

(1%

+ PtL

(20)

+ PtL)t = PtL + LP = M

(21)

dsldt

xo ,

s s

s(t)

so.

x

>

=

0.

s

s(x) s

0,

so

x

go

s

-

so.

x

go

= 0.

x

111, xtx

=

bound x(t)

xtLx >X,>O xtx

L

A,

xtMx xtx
M.

pm

< /L , , , X<~XPm -S

ds = X ~ M X dt

Am

k,

=

-Pmlhm > 0

=

-k,s

392

XIX. STABILITY

<

+

t

M

03.

P dL/dt, t?

M

(20)

bound

e-*mts,,

bound (23)

pvr ,

(25). s(x).

up

s(t)

L, s(x).

no

As s(x)

reduced energy As by

W,

0, p, 7,

K

on p, 7,

s = XtX

L=I

K

6.

393

A

'

M=LP+PtL=2

p, 7,

'")

-7

-/.L7-K2>0

/.L-7
If

(-jK

K

on

< -%,

/.L - 7 E,

- K 2 . 2 60

6,

P 6. A METRIC AS L Y A P U N O V MATRIX

K. K

(20),

M,

K xtKx

M

= KP

PI = *(P P, = +(P

+PtK

+ K-lPtK) - K-lPtK)

P,

P,

KP1

-

PItK

=0

KP,

+ P,tK

=0

P

= P,

+ P, (28),

M

= KP,

+ P,tK

= 2KP,

394

XIX. STABILITY

KP, K

proper

P,

KP, , (xtKx)

P,

7. CONCLUSIONS

P no

so

on

on no

Exercises

on a, b,

1. x4

c,

d

+ ax3 + bx2 + cx + d = 0

395

EXERCISES

2. (a)

x3 X*

a

+ ax2 + x + a = 0 + + ( b + 1) x2 + ax + b = 0 (1x3

b x2

+ 1.

on

3.

do

=

a

b

(;

3

by

References and Recommended Texts

Chapter I-VIII

11,

The Theory of Matrices, 1959,

by

book. Introduction to Matrix Analysis, book

1960.

book

Linear Algebra and Group Theory, by 1961.

book

Principles and Techniques of Applied Mathematics, 1956. on 1952,

.

Linear Algebra and Matrix Theory, good 3%

REFERENCES A N D RECOMMENDED TEXTS

397

Finite Dimensional Vector Spaces, 1958. book

1960,

Coupled Mode and Parametric Electronics, on book Introduction to Quantum Mechanics, 1960. good

1

Quantum Mechanics 1961 1962.

2,

by

Computational Methods of Linear Algebra, 1959,

V. C.

by on

on Chapter I X

Proc. I.R.E. 49, 32,

Appl. Phys. 31, 33, by

1151,

Phys. Rev. 123 Phys. Rev. 123,

by

Relativity: The Special Theory, 1960. Chapter X

Methods of Theoretical Physics, 1953.

398

REFERENCES A N D RECOMMENDED TEXTS

loc. cit. loc. cit., Chapter X I

loc cit. loc. cit. Chapter XI1

loc. cit., by Spectrum 1,

by Chapter X l l l

loc. cit. Group Theory and Its Application to the Quantum 1959. Mechanics of Atomic Spectra, Lie Algebras, 1962. Continuous Groups of Transformations, 1961. The Structure of Semisimple Lie Algebras, by 1958, 203673. The Classical Groups, Their Invariants and Representations, 1946.

Chapter XIV

loc. cit. loc. cit.

399 Chapter XV Zoc. cit. Proc. IEEE 51,

Chapter XVI

by

J. Appl. Phys. 31, Chapter XVll on Zoc. cit.,

Methods of Mathematical Physics, 1953. Zoc. cit. Matrix Iterative Analysis, 1962. Chapter XVlll loc. cit.

Zoc. cit. Chapter X I X on Zoc.

cit.,

with Applications,

Zoc.

cit.,

Stability by Liapunov’s Direct Method 1961.

SUBJECT INDEX 4

66 66

26 S, 200

5 98, 223 305 87 152 305

2 62 279 307

64 9

22

200 1 18

23, 300 302 294, 301

221

1 18 71 45, 46 142

196 198

216 56 5 5 , 215 296

73 46

222 62

349

4, 116 26

298 82 106, 203

9 88 27 26, 121

356 193, 258 102

163

50

72 135

68 74

96 293 167, 305

366 188 47 218 191

148 14 389 31

141 114 77 78 90

1 72 69

9 322

40 1

SUBJECT INDEX

349

193 349

61 21 2

101, 295 296 40, 101

25 280 4 40

373 87

356 100

240 109

E , , 52 fijk

, 349 200, 267 66, 67 306 A, 156 73 73 66, 67 306 74 A, 77 75 88 4 21 2 219 1 18

352 351 51 343 138 386 389 21 9 13, 260 241

50 47 61 47 298 188

242 on on

23 10, 57

305

44,217

327

52

108

217 49

201 45

1

173 174 109 262 212

8 340 338

51 380

105

64

32 30

372 27, 121 32 77

231 31 114, 277 198

90

GL(n), 296

138

403 69

173 391 301,305 174

366 135, 210 152

336 179 5 5 , 128, 216

140 139 135,

1

312 11, 124 241

12, 17 375 371 174 bound on

188

178 bound on, 182 1 1 1, 224 233

K

185

13 27

I 1 1, 224

K

26 188

233

58

72

64 56, 219, 221 58

102 293 14

58, 233 11 11

14 1 18 50, 100

224

224

48

K

233 56 322 325 327

386 114, 277 22 27 173 30, 173 114, 277 366, 370

L ( n ) , 302

1 1 1, 224

336 3 19

14

44

375 179

294 300

19, 21 16 17 25, 242 179, 379 50 100

301 390

11 9 10 7 9 46

22 101

404

SUBJECT INDEX

121, 124 15 128

50

379 102

148 16

3

379

61 50

100

15 195 379

222 10 86, 281

26 387 16

O(n), 296

27 76 117 30

305 97, 110, 223 by, 195 349 113, 260 113, 242 102

100 101 of, 157 386 52, 101 148 375 26 18 33 100 33 26

64,104, 287 56 by by 58 239 55

222

55

350

130

51

21 6 295 124, 127

233 233 142 196 198 S , 202

380 247 70 6

345

n

331 189 189

171 142

73 62 69 n

113, 242

a

196 198 72 307 212 198 139

405

SUBJECT INDEX

142 S , 202

296 26

196 198

4 74 50

130 379 138 85 374

4

273 4 22 55

387 387

99 322

2 100

100

85

19, 21

16 17

6

28

47, 312 281

96 282

222

on 287 259 275 on on

105

40, 241 76 39 124

T,284

T,284

117 16

50

64

66 66

73 218 216

SL(74, 296 62

51 201

193 21 9 52

91 373

51 62

281

86 281 282

96 297

72

88 87

43 45 340 218

9 379

336 195

157 86 55

124126 by 126, 127

28 386 383, 386

406

SUBJECT INDEX

148 100

371 300

155

366 367

110 383, 386

10

24 106, 203 193 262 72, 105

1 4 of, 5 200 222 148

124 250 148

66 48 44, 217 56, 219 9

380 43 86

6 122 1

56 374

18

127

17 6

375 148, 153, 159, 168,

87 222

3

17, 346

5

187

47, 312 18

2

2 47

189

219 124

U(n),296

v,

I

45

351