Matrices whose powers are $M$-matrices or $Z$-matrices

Abstract:

A matrix $A$ all of whose (positive) powers are $Z$-matrices is called here a $ZM$-matrix. A matrix is called a $ZMA$-matrix if all powers of $A$ are irreducible $Z$-matrices. We prove that the spectrum of a $ZMA$-matrix is real and only the eigenvalue minimal in absolute value may be negative. By means of an operation called inflation which generalizes the Kronecker product of two matrices, we determine the class of $ZMA$-matrices of order $n$ in terms of the classes of $ZMA$-matrices of smaller orders. We use this result to show that a $ZMA$-matrix is positively diagonally similar to a symmetric matrix. Similar results hold for $MMA$-matrices which are defined in analogy with $ZMA$-matrices in terms of $M$-matrices, and for $ZMO$-matrices which are defined to be $ZM$-matrices such that all odd powers are irreducible and all even powers reducible. We also prove that a matrix is a $ZMA$-, $ZMO$- or $MMA$-matrix under apparently weaker conditions. If $A$ is a real matrix such that all sufficiently large powers of $A$ are $Z$-matrices, then $A$ is a $ZMA$-matrix if ${A^2}$ is irreducible, $A$ is a $ZMO$-matrix if $A$ is irreducible and ${A^2}$ is reducible, and $A$ is an $MMA$-matrix if $A$ is an irreducible $Z$-matrix and some odd power of $A$ is an $M$-matrix.