On the Consistency of the Matrix Equation $$X^\top A X=B$$ when B is Symmetric

Abstract

We provide necessary and sufficient conditions for the matrix equation \(X^\top A X=B\) to be consistent when B is a symmetric matrix, for all matrices A with a few exceptions. The matrices A, B, and X (unknown) are matrices with complex entries. We first see that we can restrict ourselves to the case where A and B are given in canonical form for congruence and, then, we address the equation with A and B in such form. The characterization strongly depends on the canonical form for congruence of A. The problem we solve is equivalent to: given a complex bilinear form (represented by A) find the maximum dimension of a subspace such that the restriction of the bilinear form to this subspace is a symmetric non-degenerate bilinear form.