## 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 (u_{1}, ..., u_{n}) = G (φ{symbol}_{i1} u_{1}, ..., φ{symbol}_{in} u_{n}). for all u_{1}, ..., u_{n} ∈ U and each reordering i_{1}, ..., i_{n} 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 = F_{1} ⊕ ⋯ ⊕ F_{s} ⊕ 0, G = G_{1} ⊕ ⋯ ⊕ G_{r} ⊕ 0, and all summands F_{i} and G_{j} are nonzero and direct-sum-indecomposable, then s = r and, after a suitable reindexing, F_{i} is congruent to ±G_{i}.

Original language | English |
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Pages (from-to) | 751-762 |

Number of pages | 12 |

Journal | Linear Algebra and Its Applications |

Volume | 418 |

Issue number | 2-3 |

DOIs | |

State | Published - 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