2022/03/14 by Borobia, Alberto, Canogar, Roberto, De Terán, Fernando
#15A21 #15A24 #15A63 #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2203.07100
Given a bilinear form on \mathbb Cn, represented by a matrix A∈\mathbb Cn× n, the problem of finding the largest dimension of a subspace of \mathbb 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^\top AX=B is consistent (here X^\top 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^\top AX=B to be consistent when B is a skew-symmetric matrix. This condition is valid for most matrices A∈\mathbb 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.