On the largest eigenvalue of a symmetric nonnegative tensor

Guanglu Zhou, Liqun Qi, Soon Yi Wu

Research output: Journal article publicationJournal articleAcademic researchpeer-review

20 Citations (Scopus)

Abstract

In this paper, some important spectral characterizations of symmetric nonnegative tensors are analyzed. In particular, it is shown that a symmetric nonnegative tensor has the following properties: (i) its spectral radius is zero if and only if it is a zero tensor; (ii) it is weakly irreducible (respectively, irreducible) if and only if it has a unique positive (respectively, nonnegative) eigenvalue-eigenvector; (iii) the minimax theorem is satisfied without requiring the weak irreducibility condition; and (iv) if it is weakly reducible, then it can be decomposed into some weakly irreducible tensors. In addition, the problem of finding the largest eigenvalue of a symmetric nonnegative tensor is shown to be equivalent to finding the global solution of a convex optimization problem. Subsequently, algorithmic aspects for computing the largest eigenvalue of symmetric nonnegative tensors are discussed.
Original languageEnglish
Pages (from-to)913-928
Number of pages16
JournalNumerical Linear Algebra with Applications
Volume20
Issue number6
DOIs
Publication statusPublished - 1 Dec 2013

Keywords

  • Algorithm
  • Convergence
  • Convex optimization
  • Eigenvalue
  • Symmetric tensor

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Applied Mathematics

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