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Suppose I have a special block, Hermitian matrix

$H = \begin{bmatrix} A & B \\ B^* & A^* \end{bmatrix}$

where $*$ denotes conjugate transpose. The blocks $A$ and $B$ are themself Hermitian in this case. Are there any theorems considering the eigenvalues and eigenvectors for this special matrix?

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    @Marti$n$: Hi, and welcome to math.SE! Users with any number of "reputation points" [can comment on their own questions and answers](http://meta.stackexchange.com/questions/19756) (once you obtain 5$0$ poi$n$ts, you gain the ability to comment anywhere), but you were not able to comment because you were not signed into the account that asked the question. I've now merged your duplicate account into the original.2012-06-25

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Since in the comment, you assumed $A$ and $B$ hermitian, we can compute the characteristic polynomial $\det(H-X_{2n})$. Add to the column $k$ the column $n+k$ for $1\leq k\leq n$ to see that $\det(H-XI_{2n})=\det(A+B-XI_n)\det\pmatrix{I_n&B\\ I_n&A-XI_n}.$ Then do $R_{n+k}\leftarrow R_{n+k}-R_k$, $1\leq k\leq n$, which gives $\det(H-XI_{2n})=\det(A+B-XI_n)\det(A-B-XI_n).$ So the spectrum of $H$ is the union of the spectra of $A+B$ and $A-B$.

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    The multiplication is correct in reversed order. How do you obtain det \pmatrix{A+B-XI_n & B\\ A+B-XI_n&A-XI_n} = det \pmatrix{A+B-XI_n&B\\ A^*+B^*-XI_n&A^*-XI_n} ?2012-08-07