M-Tensors and some applications

Liping Zhang, Liqun Qi, Guanglu Zhou

Research output: Journal article publicationJournal articleAcademic researchpeer-review

202 Citations (Scopus)


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.
Original languageEnglish
Pages (from-to)437-452
Number of pages16
JournalSIAM Journal on Matrix Analysis and Applications
Issue number2
Publication statusPublished - 1 Jan 2014


  • M-tensors
  • Multivariate form
  • Positive definiteness
  • Z-tensors

ASJC Scopus subject areas

  • Analysis


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