Pairs of alternating forms and products of two skew-symmetric matrices

AbstractLet k be a field, and let S,T,S1,T1 be skew-symmetric matrices over k with S,S1 both nonsingular (if k has characteristic 2, a skew-symmetric matrix is a symmetric one with zero diagonal). It is shown that there exists a nonsingular matrix P over k with P'SP = S1, P'TP = T1 (where P' denotes the transpose of P) if and only if S-1T and S-11T1 are similar. It is also shown that a 2m×2m matrix over k can be factored as ST, with S,T skew-symmetric and S nonsingular, if and only if A is similar to a matrix direct sum B⊕B where B is an m×m matrix over k. This is equivalent to saying that all elementary divisors of A occur with even multiplicity. An extension of this result giving necessary and sufficient conditions for a square matrix to be so expressible, without assuming that either S or T is nonsingular, is included