Re: Matrix (Neumann) Convergence
- From: Michael1 <kuratowskij@xxxxxxx>
- Date: Tue, 08 May 2007 17:02:45 EDT
Hello,
I trying to prove the following Neumann Series
(I + ST + (ST)^2 + (ST)^3 + ... ) = (I - ST)^(-1)
where I is the n X n identify matrix, S is a n X n
diagonal matrix,
and T is a n X n complex symmetric matrix. Note that
(ST)^2 = (ST)
(ST), (ST)^3 = (ST)(ST)(ST), etc. and (I - ST)^(-1)
is the inverse of
(I-ST). Let s_k denote the kth diagonal element of S.
We know that
magnitude of sum_1^n s_k is less than or equal to 1
(|sum_(k=1)^n s_k|
<=1). Further, the magnitude of s_k is less than 1.
We also know that
the diagonal elements of T are zero and that
magnitude of each of the
off-diagonal elements is 1.
The series converges if the spectral radius of ST is
less than one or
if a matrix norm of ST is less than 1. Let N_m(ST)
denote the the
column norm of ST of the mth column. Then N_m(ST) =
{sum_(k=1)^n |
s_k| } - s_m. If the column norm N_m(ST) is less than
|sum_(k=1)^n
s_k| < 1 for m, then the converges. I can't prove
this statement.
Any suggestions on how I could prove if the Neunmann
series converges
based on the restrictions on S and T?
Please send a copy of your response to my e-mail
address
(bergers@xxxxxxx).
Thanks,
Scott
I didn't check the details, but an application of Gershgorin's Circle Theorem or of some stronger version of it proved in
Brualdi, R. A., Mellendorf, S., Regions in the complex plane containing the eigenvalues of a matrix. Am. Math. Mon. 101, No.10, 975-985 (1994).
could be useful.
Hope this helps
Michael
.
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