Congruence of matrix spaces, matrix tuples, and multilinear maps

Two matrix vector spaces V,W⊂Cn×n are said to be equivalent if SVR=W for some nonsingular S and R. These congruent R=ST. We prove that all matrices in V W symmetric, or skew-symmetric, then only they equivalent. Let F:U×…×U→V G:U′×…×U′→V′ symmetric skew-symmetric k-linear maps over C. If there exists a set of linear bijections φ1,…,φk:U→U′ ψ:V→V′ transforms F G, such with φ1=…=φk.