Congruence of matrix spaces, matrix tuples, and multilinear maps

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 $\varphi_1=\dots=\varphi_k$.