Please use this identifier to cite or link to this item: http://hdl.handle.net/10397/9440
Title: M-Tensors and some applications
Authors: Zhang, L
Qi, L 
Zhou, G
Keywords: M-tensors
Multivariate form
Positive definiteness
Z-tensors
Issue Date: 2014
Publisher: Society for Industrial and Applied Mathematics
Source: SIAM journal on matrix analysis and applications, 2014, v. 35, no. 2, p. 437-452 How to cite?
Journal: SIAM journal on matrix analysis and 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 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.
URI: http://hdl.handle.net/10397/9440
ISSN: 0895-4798
EISSN: 1095-7162
DOI: 10.1137/130915339
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