Prove that there exist a constant $C>0$ such that for all matrices $A\in\mathbb{C}^{3\times3}$, there exists $P\in U(3)$ and a diagonal $D\in\mathbb{C}^{3\times3}$ such that $$||P^*AP - D||^2\leq C||A^*A-AA^*||,$$ where $||B||=\sqrt{\mbox{tr}(BB^*)}$ denotes the Hilbert-Schmidt norm of $B$.
Hint: Prove the statement for the class of triangular matrices first.
Unfortunately I wasn't able to prove it even in case if $A$ is triangular. I wrote how $3\times3$ triangular matrix would look like and then tried to calculate explicitly $A^*A-AA^*$, but it doesn't look like a correct path to the solution. I also don't know how to generalize the result from the triangular case to arbitrary matrix. Are triangular matrices dense in the space of all complex matrices?
I will be grateful if someone will be able to explain the solution to this problem. Thanks in advance.