Pairs of alternating forms and products of two skew-symmetric matrices (Q762247)

Over an arbitrary field two pairs of skew-symmetric matrices S, T and \(S_ 1\), \(T_ 1\) with S and \(S_ 1\) nonsingular are shown to be equivalent by congruence iff \(S^{-1}T\) is similar to \(S_ 1^{-1}T_ 1\). This result contrasts the added criterion on the sign characteristic needed for pairs of symmetric matrices. Moreover, even-dimensional square matrices A can be written as \(A=ST\) for S and T skewsymmetric, S nonsingular iff A is similar to a block diagonal matrix of the type diag(B,B). For possible singular skew factorizations, the multiplicities of the elementary divisors of A must be even for nonzero eigenvalues and must be paired for the eigenvalue zero.