M-tensors and nonsingular M-tensors

被引：295

作者：

Ding, Weiyang ^{[1
]}

Qi, Liqun ^{[2
]}

Wei, Yimin ^{[1
,3
]}

机构：

[1] Fudan Univ, Sch Math Sci, Shanghai 200433, Peoples R China

[2] Hong Kong Polytech Univ, Dept Appl Math, Hong Kong, Hong Kong, Peoples R China

[3] Fudan Univ, Shanghai Key Lab Contemporary Appl Math, Shanghai 200433, Peoples R China

来源：

LINEAR ALGEBRA AND ITS APPLICATIONS | 2013年 / 439卷 / 10期

基金：

中国国家自然科学基金;

关键词：

M-tensors; Nonsingular M-tensors; Z-tensors; Semi-positivity; Semi-nonnegativity; H-tensors; Monotonicity; Eigenvalues; PERRON-FROBENIUS THEOREM; NONNEGATIVE TENSORS; LARGEST EIGENVALUE;

D O I：

10.1016/j.laa.2013.08.038

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

The M-matrix is an important concept in matrix theory, and has many applications. Recently, this concept has been extended to higher order tensors [18]. In this paper, we establish some important properties of M-tensors and nonsingular M-tensors. An M-tensor is a Z-tensor. We show that a Z-tensor is a nonsingular M-tensor if and only if it is semi-positive. Thus, a nonsingular M-tensor has all positive diagonal entries; an M-tensor, regarding as the limit of a sequence of nonsingular M-tensors, has all nonnegative diagonal entries. We introduce even-order monotone tensors and present their spectral properties. In matrix theory, a Z-matrix is a nonsingular M-matrix if and only if it is monotone. This is no longer true in the case of higher order tensors. We show that an even-order monotone Z-tensor is an even-order nonsingular M-tensor, but not vice versa. An example of an even-order nontrivial monotone Z-tensor is also given. (C) 2013 Elsevier Inc. All rights reserved.

引用

页码：3264 / 3278

页数：15

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