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Consider the $(m+n) \times (m+n)$ block matrix

$$M = \begin{bmatrix} A & B \\ C & D \end{bmatrix}$$

I need references where they are talking about the relation between the eigenvalues of $M$ and the eigenvalues of $A$ and $D$. I want to learn under what circumstances such a relation exists. The simplest of such a possibility is $B=0$ or $C=0$. I believe there are other non-trivial cases. Kindly share some references. Thank you.

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    $\begingroup$ suppose $A=0=D$; then still $M$ can have any set of eigenvalues (symmetrically arranged around $0$), so how can there be any relationship between the eigenvalues of $M$ and those of $A,D$ ? $\endgroup$
    – Carlo Beenakker
    Commented Jul 29, 2020 at 14:06
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    $\begingroup$ @CarloBeenakker, Thank you. In general, there need not be any relations. I want to know any special circumstances under which such a relation holds. Some papers where they talk about similar questions. This is what I am looking for. $\endgroup$
    – GA316
    Commented Jul 29, 2020 at 14:13

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To see what you might expect for a relation, consider the case of a $2\times 2$ matrix $M=\begin{pmatrix}a&b\\ c&d\end{pmatrix}$, with eigenvalues $\lambda_\pm=\tfrac{1}{2}(a+d)\pm\sqrt{4bc+(a-d)^2}$. Knowledge of $a$ and $d$ is a constraint on the sum of the eigenvalues (the trace of $M$), but there is no other constraint. I'm pretty sure this carries over to the higher dimensional case: knowledge of the matrices $A$ and $D$ fixes the trace of $M$, and hence the sum of the eigenvalues, but if you are free to vary $B,C$ there is no other constraint.

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