commuting normal matrices are simultaneously diagonalizable

All matrices in the below are complex $n\times n$ matrices.
Let $A$,$B$ be normal matrices, $AB=BA$. Then there exists a unitary matrix $Q$ such that

$Q^{H}AQ=D_1$ , $Q^{H}BQ=D_2$

where ${}^H$ is the conjugate transpose and $D_1,D_2$ are diagonal matrices.