From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik)
Subject: Re: Matrix Norm needed
Date: 20 Mar 2000 10:05:06 -0500
Newsgroups: sci.math.num-analysis
Summary: [missing]

In article <8b59q6$i6r$1@okapi.ict.pwr.wroc.pl>,
Artur Gramacki <A.Gramacki@iie.pz.zgora.pl> wrote:
:I am looking for a "specificity"  of a MATRIX NORM which guarantee
:that its value is ALWAYS <1 for any matrix with ALL eigenvalues INSIDE
:the UNIT CIRCLE.
:
:If we use the classical 2-norm, the estimation r(A) <= norm(A), where
:'r' is the spectral radius of matrix A, is too bad for the problem
:which I am solving. 
:
:Thanks in advance,
:Artur

In dimensions 2 and higher, this is impossible. To put it more
emphatically:

 to every algebra norm ||.|| (submultiplicative and unital) in the space
 of real or complex matrices n-by-n with n>1, and to every positive number
 K, there exists a matrix M such that r(M)<1 and ||M|| > K.

Actually, M can be a sufficiently large multiple of the n-by-n nilpotent
Jordan cell J (how large, depends on the norm). If nilpotent matrices are
to be rejected, take (1/2)*I + m*J (m sufficiently large).

Consolation: If r(A)<1 then there exists a positive integer p (depending
on A and on the norm) such that ||A^p|| < 1; by Gelfand's Formula,

  lim[p->infinity] ||A^p||^(1/p) = r(A)

(any norm), but you probably knew this already.

 Also, to every matrix A and every eps>0 there exists a norm ||.|| induced
by a vector norm, such that ||A|| < r(A)+eps (various authors, including
Krasnosel'skii, and Gian-Carlo Rota who did it for a "similarity norm" 

 ||A||_R = ||R * A * R^(-1)|| 

with a positive definite R and the norm induced by the Euclidean norm).

Wish I could help you, ZVK(Slavek).