Congruence of multilinear forms

Let. F : U × ⋯ × U → K, G : V × ⋯ × V → K. be two n-linear forms with n ≥ 2 on finite dimensional vector spaces U and V over a field K. We say that F and G are symmetrically equivalent if there exist linear bijections φ{symbol}₁, ..., φ{symbol}_n : U → V such that. F (u₁, ..., u_n) = G (φ{symbol}_i1 u₁, ..., φ{symbol}_in u_n). for all u₁, ..., u_n ∈ U and each reordering i₁, ..., i_n of 1, ..., n. The forms are said to be congruent if φ{symbol}₁ = ⋯ = φ{symbol}_n. Let F and G be symmetrically equivalent. We prove that. (i)if K = C, then F and G are congruent;(ii)if K = R, F = F₁ ⊕ ⋯ ⊕ F_s ⊕ 0, G = G₁ ⊕ ⋯ ⊕ G_r ⊕ 0, and all summands F_i and G_j are nonzero and direct-sum-indecomposable, then s = r and, after a suitable reindexing, F_i is congruent to ±G_i.