ABSTRACT
Given a bilinear form on , represented by a matrix
, the problem of finding the largest dimension of a subspace of
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
is consistent (here
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
to be consistent when B is a skew-symmetric matrix. This condition is valid for most matrices
. 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.
Disclosure statement
No potential conflict of interest was reported by the author(s).