Recently I've been learning *congruence on complex matrix. The definition of *congruence is that:
Let $A, B \in \mathsf{M}_n$. $A$ and $B$ are *-congruent if and only if there exists $S \in \mathsf{M}_n$ such that $A = SBS^\ast$.
A few notions come into view: There are papers studying simultaneous diagonalizability or similarity and canonical form (R.Horn) under *-congruence.
The techniques make sense and the paper above makes sense too. I found some places in the paper and show very interesting geometry interpretation. I wonder how we can interpret *-congruence as a whole. When we have the normal notion of similarity, we kind of know that if $A = SBS^{-1}$, $A$ is the result of $B$ under the change of basis of column vector of $S$. Do we get a similar interpretation for *-congruence?
