Congruence and $A^{-1}{A}^{\ast }$

A canonical form under congruence for nonsingular complex matrices may be deduced from the canonical pair form for Hermitian matrices in a straightforward way. This allows the study of $A^{-1}{A}^{\ast }$ and what it says about the congruence class of $A$. In turn, alternate congruential forms may be given along with a further generalization of Sylvester's law of inertia and an application to C-S equivalence.