M-tensors and nonsingular M-tensors

Weiyang Ding, Liqun Qi, Yimin Wei

Research output: Journal article publicationJournal articleAcademic researchpeer-review

163 Citations (Scopus)


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.
Original languageEnglish
Pages (from-to)3264-3278
Number of pages15
JournalLinear Algebra and Its Applications
Issue number10
Publication statusPublished - 15 Nov 2013


  • Eigenvalues
  • H-tensors
  • M-tensors
  • Monotonicity
  • Nonsingular M-tensors
  • Semi-nonnegativity
  • Semi-positivity
  • Z-tensors

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Discrete Mathematics and Combinatorics
  • Geometry and Topology
  • Numerical Analysis

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