Eigenvalues and invariants of tensors

Liqun Qi

Research output: Journal article publicationJournal articleAcademic researchpeer-review

149 Citations (Scopus)

Abstract

A tensor is represented by a supermatrix under a co-ordinate system. In this paper, we define E-eigenvalues and E-eigenvectors for tensors and supermatrices. By the resultant theory, we define the E-characteristic polynomial of a tensor. An E-eigenvalue of a tensor is a root of the E-characteristic polynomial. In the regular case, a complex number is an E-eigenvalue if and only if it is a root of the E-characteristic polynomial. We convert the E-characteristic polynomial of a tensor to a monic polynomial and show that the coefficients of that monic polynomial are invariants of that tensor, i.e., they are invariant under co-ordinate system changes. We call them principal invariants of that tensor. The maximum number of principal invariants of mth order n-dimensional tensors is a function of m and n. We denote it by d (m, n) and show that d (1, n) = 1, d (2, n) = n, d (m, 2) = m for m ≥ 3 and d (m, n) ≤ mn - 1+ ⋯ + m for m, n ≥ 3. We also define the rank of a tensor. All real eigenvectors associated with nonzero E-eigenvalues are in a subspace with dimension equal to its rank.
Original languageEnglish
Pages (from-to)1363-1377
Number of pages15
JournalJournal of Mathematical Analysis and Applications
Volume325
Issue number2
DOIs
Publication statusPublished - 15 Jan 2007

Keywords

  • Eigenvalue
  • Invariant
  • Rank
  • Supermatrix
  • Tensor

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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