Inverses of M-type matrices created with irreducible eventually nonnegative matrices

An M-matrix is a matrix that can be expressed as alpha I - P, where P is entry wise nonnegative and alpha >= rho(P). It is well known that the inverse of a nonsingular irreducible M-matrix is positive. In this paper, matrices of the form alpha I - P, where P is an irreducible eventually nonnegative matrix and a > (P), are studied. It is shown that if index(0) (P) <= 1, then there exists a positive number. such that if lambda > alpha > rho (P), then the inverse of alpha I - P is positive. (c) 2006 Elsevier Inc. All rights reserved.