Diagonalizing blocks of a unitary matrix

Hi, All!

My problem:

we have a complex unitary 2n*2n matrix A. It has four n*n blocks A_11
(left upper corner), A_12 (right upper corner), A_21, A_22.

The question: in which cases the blocks can be diagonalized by the same
unitary transormation matrix, i.e. there exists a unitary n*n matrix U
such that all U^* A_ij U (ij=1,2) are diagonal? Clearly, the blocks must
commute with each other, but this does not seem to be sufficient. Are
there some criteria (in terms of the whole matrix A), etc.?

K.P.
.

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