A survey on the spectral theory of nonnegative tensors

Kungching Chang, Liqun Qi, Tan Zhang

Research output: Journal article publicationJournal articleAcademic researchpeer-review

73 Citations (Scopus)

Abstract

This is a survey paper on the recent development of the spectral theory of nonnegative tensors and its applications. After a brief review of the basic definitions on tensors, the H-eigenvalue problem and the Z-eigenvalue problem for tensors are studied separately. To the H-eigenvalue problem for nonnegative tensors, the whole Perron-Frobenius theory for nonnegative matrices is completely extended, while to the Z-eigenvalue problem, there are many distinctions and are studied carefully in details. Numerical methods are also discussed. Three kinds of applications are studied: higher order Markov chains, spectral theory of hypergraphs, and the quantum entanglement.
Original languageEnglish
Pages (from-to)891-912
Number of pages22
JournalNumerical Linear Algebra with Applications
Volume20
Issue number6
DOIs
Publication statusPublished - 1 Dec 2013

Keywords

  • Eigenvalues for tensors
  • High order Markov chain
  • Perron-Frobenius theorem

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Applied Mathematics

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