$M$-Tensors and Some Applications

Abstract

We introduce $M$-tensors. This concept extends the concept of $M$-matrices. We denote $Z$-tensors as the tensors with nonpositive off-diagonal entries. We show that $M$-tensors must be $Z$-tensors and the maximal diagonal entry must be nonnegative. The diagonal elements of a symmetric $M$-tensor must be nonnegative. A symmetric $M$-tensor is copositive. Based on the spectral theory of nonnegative tensors, we show that the minimal value of the real parts of all eigenvalues of an $M$-tensor is its smallest H$^+$-eigenvalue and also is its smallest H-eigenvalue. We show that a $Z$-tensor is an $M$-tensor if and only if all its H$^+$-eigenvalues are nonnegative. Some further spectral properties of $M$-tensors are given. We also introduce strong $M$-tensors, and some corresponding conclusions are given. In particular, we show that all $H$-eigenvalues of strong $M$-tensors are positive. We apply this property to study the positive definiteness of a class of multivariate forms associated with $Z$-tensors. We also propose an algorithm for testing the positive definiteness of such a multivariate form.