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?
Best Answer
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 $A-B$.