If A and B are conjugate then there exists an orthogonal transformation, such that:

A= CBC^t

Prove that Tr(CBC^t)=Tr(B) (by definition).

(or if you want you can use the next identity, Tr(AB)=Tr(BA), and the fact that

CC^t=I).

Now I can't recall how to show that Tr A =Tr B yields A=CBC^t...

I'll come back to it later if no one else tries to show it.

Cheers.