Inverses of Perron complements of inverse M-matrices

Abstract

The concept of the Perron complement of a nonnegative and irreducible matrix was introduced by Meyer in 1989 and it was used by him to construct an algorithm for computing the stationary distribution vector for Markov chains. Here we consider properties of the Perron complement of an $\text{n×n}$ matrix K which is an inverse of an irreducible M-matrix. We first show that the Perron complements of K are inverses of M-matrices and that the inverses of associated principal submatrices of K are sandwiched between the inverses of the Perron complements of K and the inverses of the corresponding Schur complements of K. We then investigate the directed graph of the inverse of the Perron complements of such matrices K.