Congruence of matrix spaces, matrix tuples, and multilinear maps

Two matrix vector spaces V,W⊂C^n×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×…×U→V and G:U^′×…×U^′→V^′ be symmetric or skew-symmetric k-linear maps over C. If there exists a set of linear bijections φ₁,…,φ_k:U→U^′ and ψ:V→V^′ that transforms F to G, then there exists such a set with φ₁=…=φ_k.