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M-tensors and some applications

Li-Ping Zhang Department of Mathematical Sciences, Tsinghua University Liqun Qi Department of Applied Mathematics, Hong Kong Polytechnic University Guanglu Zhou Department of Mathematics and Statistics, Curtin University, Perth, Australia

Numerical Linear Algebra mathscidoc: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.
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@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.
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