My question is:
If $A$ and $B$ are two Hermitian matrices, and $AB$ is also a Hermitian matrix, then how do prove that both $A$ and $B$ are diagonalizable through the same unitary matrix (i.e the unitary matrix that diagonalizes $A$, diagonalizes $B$ as well).
It is obvious that in order for $AB$ to be Hermitian, $A$ and $B$ have to commute, i.e: $AB=BA$. Can anyone tell me how to prove that the same unitary matrix that diagonalizes $A$, diagonalizes $B$ as well?