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This relationship is called congruence, and for symmetric (or hermitian) matrices $A, B$ such an $X$ exists if and only if the matrices have the same inertia (the same number of positive, negative, zero eigenvalues). Fro general matrices congruence is much less well known, but it is quite well understood, see:

https://gauss.uc3m.es/fteran/papers/2010-1.pdf