On the consistency of the matrix equation $$X^\top A X=B$$ when B is symmetric: the case where CFC(A) includes skew-symmetric blocks

Abstract In this paper, which is a follow-up to Borobia et al. (Mediterr J Math, 18:40, 2021), we provide necessary and sufficient condition for the matrix equation $$X^\top AX=B$$ X ⊤ A = B be consistent when B symmetric. The depends on canonical form congruence of A , proved all matrices most them. This result improves main one in previous since stronger than that reference, sufficiency guaranteed larger set (namely, those whose congruence, CFC( ), includes skew-symmetric blocks).