Journal Article

M$ -Tensors and Some Applications

Qi, Liqun, Zhang, Liping, Zhou, Guanglu

2014

Overview

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. [PUBLICATION ABSTRACT]

• Publisher: Society for Industrial and Applied Mathematics
• Subject: Algorithms / Applied mathematics / Computational mathematics / Eigenvalues / Eigenvectors / Grants / Mathematical analysis / Spectra / Spectral theory / Tensors