Symmetric nonnegative tensors and copositive tensors

Liqun Qi

Research output: Journal article publicationJournal articleAcademic researchpeer-review

141 Citations (Scopus)

Abstract

We first prove two new spectral properties for symmetric nonnegative tensors. We prove a maximum property for the largest H-eigenvalue of a symmetric nonnegative tensor, and establish some bounds for this eigenvalue via row sums of that tensor. We show that if an eigenvalue of a symmetric nonnegative tensor has a positive H-eigenvector, then this eigenvalue is the largest H-eigenvalue of that tensor. We also give a necessary and sufficient condition for this. We then introduce copositive tensors. This concept extends the concept of copositive matrices. Symmetric nonnegative tensors and positive semi-definite tensors are examples of copositive tensors. The diagonal elements of a copositive tensor must be nonnegative. We show that if each sum of a diagonal element and all the negative off-diagonal elements in the same row of a real symmetric tensor is nonnegative, then that tensor is a copositive tensor. Some further properties of copositive tensors are discussed.
Original languageEnglish
Pages (from-to)228-238
Number of pages11
JournalLinear Algebra and Its Applications
Volume439
Issue number1
DOIs
Publication statusPublished - 1 Jun 2013

Keywords

  • Copositive tensor
  • H-eigenvalue
  • Nonnegative tensor

ASJC Scopus subject areas

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

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