Congruence of matrix spaces, matrix tuples, and multilinear maps

TL;DR

This paper investigates the conditions under which matrix spaces and multilinear maps are congruent or equivalent, establishing when congruence coincides with equivalence for symmetric or skew-symmetric cases and exploring transformations of multilinear maps.

Contribution

It proves that congruence and equivalence coincide for symmetric or skew-symmetric matrix spaces and shows that transformations of multilinear maps can be simplified to identical linear bijections.

Findings

Congruence and equivalence are equivalent for symmetric or skew-symmetric matrix spaces.

Transformations of multilinear maps can be achieved with identical linear bijections.

Provides conditions under which multilinear maps are related by the same set of linear bijections.

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 $\varphi_1=\dots=\varphi_k$.