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 <Tiberiu.Rotaru@info.unine.ch> 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