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 $F:U\times \cdots \times U\to V$ and $G:U^{\prime }\times \cdots \times U^{\prime }\to V^{\prime }$ be symmetric or skew-symmetric $k$-linear maps over $\mathbb{C}$. If there exists a set of linear bijections ${\phi }_1,\dots ,{\phi }_k:U\to U^{\prime }$ and $\psi :V\to V^{\prime }$ that transforms $F$ to $G$, then there exists such a set with ${\phi }_1=\cdots ={\phi }_k$.

18 pages

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