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

TL;DR

This paper establishes a comprehensive criterion for the consistency of the matrix equation $X^\top A X=B$ with symmetric $B$, based on the canonical form of $A$, extending previous results and covering more cases.

Contribution

It provides a necessary and sufficient condition for the equation's consistency when $B$ is symmetric, especially for matrices with skew-symmetric blocks in their canonical form.

Findings

Condition depends on the canonical form for congruence of $A$

Necessary for all matrices $A$, sufficient for matrices with skew-symmetric blocks in CFC($A$)

Improves previous results by broadening the class of matrices for which sufficiency holds

Abstract

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