I have currently a problem with a 8x8 matrix and maybe someone knows a neat trick how to solve this in an elegant way.
I have an 8x8 matrix consisting of four 4x4 submatrices:
$M=\begin{pmatrix} A & B \\ B^{\dagger} & -\tilde{A} \end{pmatrix}$
So $A$ and $-\tilde{A}$ have the same structure with different values. All submatrices are hermitian and thus $M$ is also hermitian.
Now: I already know the analytic results for the diagonalization of each submatrix, i.e. I already have all $U_X$ for which I get the diagonal matrix $D_X$:
$ D_X = U_X^{\dagger} X U_X$ for $X=A,B,\tilde{A}$.
So the question: outgoing from the 4 transformations which diagonalize the submatrices, is there a smart way to construct the transformation which diagonalizes $M$?
Edit: yes, $A$,$B$ and $\tilde{A}$ are hermitian Edit2: Notation changed for clarity