# MathSciDoc: An Archive for Mathematician ∫

#### Numerical Linear Algebramathscidoc:1804.26001

SIAM Journal on Matrix Analysis and Applications, 35, (2), 437-452, 2014.4
We introduceM-tensors. This concept extends the concept ofM-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.
```@inproceedings{li-ping2014m-tensors,
title={M-tensors and some applications},
author={Li-Ping Zhang, Liqun Qi, and Guanglu Zhou},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180403164549420915023},
booktitle={SIAM Journal on Matrix Analysis and Applications},
volume={35},
number={2},
pages={437-452},
year={2014},
}
```
Li-Ping Zhang, Liqun Qi, and Guanglu Zhou. M-tensors and some applications. 2014. Vol. 35. In SIAM Journal on Matrix Analysis and Applications. pp.437-452. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180403164549420915023.