An even order symmetric B tensor is positive definite

Liqun Qi, Yisheng Song

Research output: Journal article publicationJournal articleAcademic researchpeer-review

56 Citations (Scopus)

Abstract

It is easily checkable if a given tensor is a B tensor, or a B0tensor or not. In this paper, we show that a symmetric B tensor can always be decomposed to the sum of a strictly diagonally dominated symmetric M tensor and several positive multiples of partially all one tensors, and a symmetric B0tensor can always be decomposed to the sum of a diagonally dominated symmetric M tensor and several positive multiples of partially all one tensors. When the order is even, this implies that the corresponding B tensor is positive definite, and the corresponding B0tensor is positive semi-definite. This gives a checkable sufficient condition for positive definite and semi-definite tensors. This approach is different from the approach in the literature for proving a symmetric B matrix is positive definite, as that matrix approach cannot be extended to the tensor case.
Original languageEnglish
Pages (from-to)303-312
Number of pages10
JournalLinear Algebra and Its Applications
Volume457
DOIs
Publication statusPublished - 15 Sep 2014

Keywords

  • B tensor
  • M tensor
  • Partially all one tensor
  • Positive definiteness

ASJC Scopus subject areas

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

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