The E-eigenvectors of tensors

Shenglong Hu, Liqun Qi

Research output: Journal article publicationJournal articleAcademic researchpeer-review

4 Citations (Scopus)

Abstract

We first show that the eigenvector of a tensor is well defined. The differences between the eigenvectors of a tensor and its E-eigenvectors are the eigenvectors on the nonsingular projective variety (Formula presented.). We show that a generic tensor has no eigenvectors on (Formula presented.). Actually, we show that a generic tensor has no eigenvectors on a proper nonsingular projective variety in (Formula presented.). By these facts, we show that the coefficients of the E-characteristic polynomial are algebraically dependent. Actually, a certain power of the determinant of the tensor can be expressed through the coefficients besides the constant term. Hence, a nonsingular tensor always has an E-eigenvector. When a tensor (Formula presented.) is nonsingular and symmetric, its E-eigenvectors are exactly the singular points of a class of hypersurfaces defined by (Formula presented.) and a parameter. We give explicit factorization of the discriminant of this class of hypersurfaces which completes Cartwright and Strumfels’ formula. We show that the factorization contains the determinant and the E-characteristic polynomial of the tensor (Formula presented.) as irreducible factors.
Original languageEnglish
Pages (from-to)1388-1402
Number of pages15
JournalLinear and Multilinear Algebra
Volume62
Issue number10
DOIs
Publication statusPublished - 1 Jan 2014

Keywords

  • E-characteristic polynomial
  • E-eigenvector
  • invariants
  • nonsingular
  • tensor

ASJC Scopus subject areas

  • Algebra and Number Theory

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