# Congruence of matrix spaces, matrix tuples, and multilinear maps

Research paper by **Genrich R. Belitskii, Vyacheslav Futorny, Mikhail Muzychuk, Vladimir V. Sergeichuk**

Indexed on: **30 Sep '20**Published on: **29 Sep '20**Published in: **arXiv - Mathematics - Representation Theory**

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#### Abstract

Two matrix vector spaces $V,W\subset \mathbb C^{n\times n}$ are said to be
equivalent if $SVR=W$ for some nonsingular $S$ and $R$. These spaces are
congruent if $R=S^T$. We prove that if all matrices in $V$ and $W$ are
symmetric, or all matrices in $V$ and $W$ are skew-symmetric, then $V$ and $W$
are congruent if and only if they are equivalent. Let $F: U\times\dots\times
U\to V$ and $G: U'\times\dots\times U'\to V'$ be symmetric or skew-symmetric
$k$-linear maps over $\mathbb C$. If there exists a set of linear bijections
$\varphi_1,\dots,\varphi_k:U\to U'$ and $\psi:V\to V'$ that transforms $F$ to
$G$, then there exists such a set with $\varphi_1=\dots=\varphi_k$.