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