Tuesday 27 June 2017

linear algebra - Eigenvalues of a general block hermitian matrix

I have a question regarding the eigenvalues of a block Hermitian matrix as a function of the eigenvalues of the diagonal block matrices and the off-diagonal matrices. In particular, I am interested in the 2x2 block case.



I have checked some previous posts [1]: Eigenvalues of certain block hermitian matrix and in Wikipedia, and it is clear to me that the solution for the case $M_1=\left(\begin{array}{} A & B\\ B &A \end{array}\right)$, where $M_1$, $A$ and $B$ are Hermitian, can be derived.



Nevertheless, I would like to know if it is possible, in the following case: $M_2=\left(\begin{array}{} A & B\\ B^{H} &C \end{array}\right)$ where $M_2$, $A$ and $C$ are Hermitian and $B$ corresponds to the off-diagonal block, to say something about the eigenvalues of $M_2$ as a function of the eigenvalues of $A$ and $C$ and the matrix $B$.



Best regards.

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