Solution
= Solution
Suppose $A$ is diagonalized by a unitary <matrix>:
$$
A=UDU^\dagger,
$$
where $D$ is diagonal. Then
$$
A^\dagger=UD^\dagger U^\dagger.
$$
Since the diagonal <matrices> $D$ and $D^\dagger$ commute,
$$
AA^\dagger=UDD^\dagger U^\dagger
=UD^\dagger DU^\dagger=A^\dagger A.
$$
\b[Thus $A$ is a <normal matrix>.]