9.9. The Case of Toeplitz and Circulanl Matrices which is (\1.32).
231
D
We n" w lum 10
inv~r",s
"fTo"l'lil'. "'~lr...
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9.9. The Case of Toeplitz and Circulanl Matrices which is (\1.32).
231
D
We n" w lum 10
inv~r",s
"fTo"l'lil'. "'~lri<..,,;.
The" ....m 9.2n. If I> 1m" "" .<"m.' "n T ,,,,d .... ind " == O. t"""
for caeh E: > 0.
Pro"f . The sing u l~r \al ue~ "f T.- ' (") ~re J/ a; (j == I. . . . . n). Thus. by Theorem 9.23,
"
""j • ' (......' + •• •+ "'"')
F. X. = -a , n
" 1 Xl = •
Uj
2
n(n + 2)
".
a,_ (_":' + .. . + a:"
_ ( -'- + ... + -'-) ') . .. 1 " l
.
,
II
Si""e T (I.) is inv~nibk . we can invoke fom'u la (9.30) wilh t = - 2 and lbe Ii"'t of fonn"I'b (6,16) 10deduc" th~l
A< always " l
X; s
2/ (n + 21 , we "blain f""" Le",m~ 9.24Ih~1 ...... " -= O n+2w c1
1' ( IX. - i' I ~ c ) ::: -)
('n ) -
.
o
I f Iwind hl = t ~ I . lhcn T. (") need not be invertihle for all sufficienlly large dimensi""" ". We Iheref" re c"nsider lhe: M""",-I'enro>e in\'e""" T/ (b ). wbich c"inc ilb wilh T.-' (f» in lhe ca.", or in,-cn ibilily.
Th........m'l.27. If b 1!11,\ "" .<"m.' "n T ",UJ Iw;",l bl
~
I. II!.."/<,, ..",'I! ,- > O.
Pro"f . T1J<.o.... m 9,4 1ells us Ihal if Iwind l>] = A -==- I, It>..n l singular \'~l ues of T.(b ) converge 10 zero with e~l"~lenl ial 'I'<'ed, ,,/ ::: C.. - Y' (y > 0) for ( ::: t , while the remaining singular valu~ slay a....ay rrom ""ro. a / ~ J. > 0 f"r ( ::= + I. Thus, lhe singular val""" " f T/ (I. ) are
*
,
0.. ... 0. - . . . .. - " . -
,
".
" ; +1 " ;
