$M$-Tensors and Some Applications

Abstract

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 Ztensors 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 Mtensor 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. 1. Introduction. Tensors are increasingly ubiquitous in various areas of applied, computational, and industrial mathematics and have wide applications in data analysis and mining, information science, signal/image processing, and computational biology as well [5,9,15,17]. A tensor can be regarded as a higher-order generalization of a matrix, which takes the form Key words. M -tensors, H

ℋ-tensors and nonsingular ℋ-tensors

“…Definition 2 [5,18] We call a tensor A as an M -tensor, if there exists a nonnegative tensor B and a positive real number η ρ(B) such that…”

“…Definition 5 [18] Let A be an m-order and n-dimensional tensor. A is diagonally dominant if for i ∈ [n],…”

“…They are closely related with spectral graph theory, the stationary distribution of Markov chains, and the convergence of iterative methods for singular linear equations. Zhang et al [18] extended M -matrices to M -tensors and studied their properties. Ding et al [5] gave ten equivalent conditions for M -tensors and extended the other two definitions of nonsingular M -matrices, semi-positivity and monotonicity [1,13], to higher-order tensors.…”

Criteria for Nonsingular H-tensors

“…Generally, tensor is a higher-order extension of matrix. A high order tensor is a multi-way array whose entries are addressed via multiple indices in the following form: In paper [2] and [4], the authors gave some properties and applications of M-tensors. In paper [3], the authors gave some properties of H-tensors.…”

New criteria for H-tensors and an application

“…It is known that the positive definiteness of multivariate polynomial f .x/ plays an important role in the stability study of nonlinear autonomous systems [9,10]. However, for n 3 and m 4, it is not easy to identify the positive definiteness of such a multivariate form.…”