Ray pattern matrices requiring Perron–Frobenius properties

Abstract

A ray pattern matrix is a matrix whose nonzero entries are all unimodular. An n×n ray pattern matrix A is said to require the weak Perron–Frobenius property if for any n×n entrywise positive matrix K, the spectral radius ρ(K∘A) is one of the eigenvalues of K∘A; to require the strong Perron–Frobenius property if ρ(K∘A) is an algebraically simple eigenvalue of K∘A and the left and right eigenvectors associated to ρ(K∘A) are strictly nonzero. In this paper, necessary and sufficient conditions for ray pattern matrices requiring the weak/strong Perron–Frobenius property are given.