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.