Unitarity similarity transformation

Question

Prove that $R$ and $R^{\dagger}$ can be diagonalized by a common unitary similarity transformation if $R^{\dagger}$ is commutable with $R$.

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Let $R = SMS^{-1}$, where $M$ is diagonal and $S$ is unitary.

$$R^{\dagger} = (SMS^{-1})^{\dagger} = SM^{\dagger}S^{-1}$$

since $SS^{\dagger}=I$.

My point is, I have shown that $R$ and $R^\dagger$ can both be diagonalized by the same unitary matrix/transformation without to need to use the commutation information.

Could it be that i have committed some assumption that isn't necessarily true without I realized that?

Answer 1

Your answer is correct, and I think that it is complete. That said, to others' taste, it might be that you haven't given enough justification for each step.

To expand your work to make things a bit more explicit: if we have $R = SMS^{-1}$ with $M$ diagonal and $S$ unitary, then $$ R^\dagger = (SMS^{-1})^\dagger = (SMS^\dagger)^\dagger = S^{\dagger \dagger} M^\dagger S^\dagger = S M^\dagger S^{-1}. $$ Because $M$ is diagonal, $M^\dagger$ must also be diagonal. Thus, the equation $R^\dagger = SM^\dagger S^{-1}$ is a diagonalization of $R^\dagger$ using the same similarity. The conclusion follows.

Answer 2

I think the more difficult part of the question is proving that $R$ can even be diagonalized by a unitary transformation. This is true since $R$ is normal, but it isn't obvious.