From: israel@math.ubc.ca (Robert Israel)
Subject: Re: Perron-Frobenius
Date: 7 Dec 1999 10:00:02 -0600
Newsgroups: sci.math.research
Keywords: extensions to nonnegative matrices
In article <384bdd99.0@sitelnet.unine.ch>,
Tiberiu Rotaru writes:
> In what cases the Perron-Frobenius theorem
> remains valid for irreducibile, primitive but not nonnegative
> matrices?
The most obvious extension would be: if A is a matrix such that
for some positive integer k, all entries of A^k are (strictly) positive,
then we can apply Perron-Frobenius to A^k. Then A has a simple eigenvalue
lambda such that lambda^k > 0; all other eigenvalues are smaller in
absolute value; A and A^T have eigenvectors u and v for eigenvalue lambda
which are strictly positive; and lambda^(-n) A^n x -> (v^T x) u as n -> infinity.
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2