The equation XTAX=B with B skew-symmetric: how much of a bilinear form is skew-symmetric?

Abstract

Given a bilinear form on Cn, represented by a matrix A∈Cn×n, the problem of finding the largest dimension of a subspace of Cn such that the restriction of A to this subspace is a non-degenerate skew-symmetric bilinear form is equivalent to finding the size of the largest invertible skew-symmetric matrix B such that the equation X⊤AX=B is consistent (here X⊤ denotes the transpose of the matrix X). In this paper, we provide a characterization, by means of a necessary and sufficient condition, for the matrix equation X⊤AX=B to be consistent when B is a skew-symmetric matrix. This condition is valid for most matrices A∈Cn×n. To be precise, the condition depends on the canonical form for congruence (CFC) of the matrix A, which is a direct sum of blocks of three types. The condition is valid for all matrices A except those whose CFC contains blocks, of one of the types, with size smaller than 3. However, we show that the condition is necessary for all matrices A.