Show that a matrix is diagonal

Question

My task given by my professor was the following:

$Let \,\, A, \,\, B \in \mathbb{C}^{n\times n}$ be selfadjoint and such that $[A, B] := AB-BA=0.$ Show that $C:= A+iB$ is normal. Show further that there is a unitary $U \in \mathbb{C}^{n\times n} \; \text{such that} \; U^*AU \; \text{and} \; U^*BU$ are both diagonal.

I have proven that $C$ is normal but I'm having problems with proving that the terms are both diagonal matricies.

Answer 1

Hint: If $U^* C U = D$ is diagonal, $U^* C^* U = (U^* C U)^* = D^*$ which is also diagonal. Now what are $(C + C^*)/2$ and $(C - C^*)/(2i)$?