Today at lesson, the professor brought up the notion of unitary similarity when talking about Schur theorem.
After lesson, we talked about complete characterizations of this similarity (not only for normal matrices, where it is easy). Turned out we didn't know any.
We know that two unitarily similar matrices must have
- same Jordan Form (since they must be similar)
- same singular values
- similar real and imaginary part (where the real part of a matrix $A$ is $A+A^*/2$ and the imaginary part $A-A^*/2\text{i}$)
Now I found written in a forum that the first two conditions are sufficient, but I can't find a proof or a counterexample, or a reference to this.
Actually the exact lemma cited was
$A,B$ are unitarily similar iff they are similar and $ tr(A^*A)=tr(B^*B)$
Any help?