Congruence of multilinear forms

Genrich R. Belitskii, Vladimir V. Sergeichuk

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

Let. F : U × ⋯ × U → K, G : V × ⋯ × V → K. be two n-linear forms with n ≥ 2 on finite dimensional vector spaces U and V over a field K. We say that F and G are symmetrically equivalent if there exist linear bijections φ{symbol}1, ..., φ{symbol}n : U → V such that. F (u1, ..., un) = G (φ{symbol}i1 u1, ..., φ{symbol}in un). for all u1, ..., un ∈ U and each reordering i1, ..., in of 1, ..., n. The forms are said to be congruent if φ{symbol}1 = ⋯ = φ{symbol}n. Let F and G be symmetrically equivalent. We prove that. (i)if K = C, then F and G are congruent;(ii)if K = R, F = F1 ⊕ ⋯ ⊕ Fs ⊕ 0, G = G1 ⊕ ⋯ ⊕ Gr ⊕ 0, and all summands Fi and Gj are nonzero and direct-sum-indecomposable, then s = r and, after a suitable reindexing, Fi is congruent to ±Gi.

Original languageEnglish
Pages (from-to)751-762
Number of pages12
JournalLinear Algebra and Its Applications
Volume418
Issue number2-3
DOIs
StatePublished - 15 Oct 2006

Keywords

  • Equivalence and congruence
  • Multilinear forms
  • Tensors

ASJC Scopus subject areas

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

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