Max Trace of U(N) Matrices: A Complex Problem

[DavidK]

Assume $$A$$ is a complex $$N \times N$$ matrix. It is well known that $$\max_{V \in U(N)} |Tr(AV)| = Tr(\sqrt{AA^{\dagger}})$$. But what is
$$\max_{V \in U(N)} Re(Tr(AV))$$?

$$U(N)$$ is the group of unitary $$N \times N$$ matrices.

(I could not preview my post properly, so I apologize for any latex-misstakes )

 

[DavidK]

You may ignore this question. It was not very difficult to show that $$\max_{V \in U(N)} Re(Tr(AV))=Tr(\sqrt{AA^{\dagger}})$$ aswell.