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.

Title	commuting normal matrices are simultaneously diagonalizable
Canonical name	CommutingNormalMatricesAreSimultaneouslyDiagonalizable
Date of creation	2013-03-22 15:26:51
Last modified on	2013-03-22 15:26:51
Owner	georgiosl (7242)
Last modified by	georgiosl (7242)
Numerical id	7
Author	georgiosl (7242)
Entry type	Corollary
Classification	msc 15A23