Pseudo-spectra theory of tensors and tensor polynomial eigenvalue problems

Maolin Che, Guoyin Li, Liqun Qi, Yimin Wei

Research output: Journal article publicationJournal articleAcademic researchpeer-review

4 Citations (Scopus)

Abstract

This paper is devoted to the extension of the ϵ-pseudo-spectra theory from matrices to tensors. Based on the definition of an eigenpair of real symmetric tensors and results on the ϵ-pseudo-spectrum of square matrices, we first introduce the ϵ-pseudo-spectrum of a complex tensor and investigate its fundamental properties, such as its computational interpretations and the link with the stability of its related homogeneous dynamical system. We then extend the ϵ-pseudo-spectrum to the setting of tensor polynomial eigenvalue problems. We further derive basic structure of the ϵ-pseudo-spectrum for tensor polynomial eigenvalue problems including the symmetry, boundedness and number of connected components under suitable mild assumptions. Finally, we discuss the implication of the ϵ-pseudo-spectrum for computing the backward errors and the distance from a regular tensor polynomial to the nearest irregular tensor polynomial.
Original languageEnglish
Pages (from-to)536-572
Number of pages37
JournalLinear Algebra and Its Applications
Volume533
DOIs
Publication statusPublished - 15 Nov 2017

Keywords

  • Backward error
  • Determinant
  • Eigenpair
  • Nonnegative weights
  • Regular tensor polynomial
  • Tensor polynomial eigenvalue problems
  • Tensors
  • ϵ-pseudo-spectrum

ASJC Scopus subject areas

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

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