On relationships between several classes of Z-matrices, M-matrices and nonnegative matrices

Abstract

In this paper we establish relationships between several classes of well-known matrices and some new classes of matrices. We consider matrices whose powers are irreducible Z-matrices or M-matrices. These matrices are called ZMA-matrices or MMA-matrices. We establish a relationship between theses classe of matrices and the class of Soules matrices. Moreover, we prove that symmetric ZMA-matrices and MMA-matrices are diagonally congruent to matrices that we call Z-SUMs and M-SUMs. Conversely, Z-SUMs and M-SUMs are diagonally congruent to ZMA-matrices and MMA-matrices. This relationship exploits the structure of ZMA-matrices and MMA-matrices and several well-known properties of ZMA-matrices follow immediately. Z-SUMs and M-SUMs are closely related to strictly ultrametric matrices (SUMs). We also characterize inverse ZMA-matrices. We prove that symmetric inverse ZMA-matrices are diagonally congruent to shifted SUMs and that shifted SUMs are diagonally congruent to symmetric inverse ZMA-matrices.