I have the next block matrix

$$ M = \begin{bmatrix}A & B \\ C &D\end{bmatrix}, $$ where $A, D$ are Hurwitz (eigenvalues with negative real part) square matrices of different dimensions and $B, C$ have the right dimensions. Furthermore, we have the next relations $B D^{-1} C = 0$, $BC = 0$ and $CB=0$, where $0$ stands for the appropriated zero matrix.

I know that $\det(M) = \det(D) \det(A-B D^{-1} C)=\det(D) \det(A)$. Can I say something about the eigenvalues of $M$ given this special structure?

I guess this is the right characteristic polynomial right? $\lambda(M) = \det(D-\lambda I)\det((A-\lambda I)-B (D-\lambda I)^{-1} C)$ . I do not know if $B (D-\lambda I)^{-1} C$ could be nicely approximated, simplified or considered as a perturbation (small) matrix?

And what about the next special case, if $A, B, C, D$ are square matrices with the same dimensions, can be simplified in some way the characteristic polynomial? i.e. is true for such case that

$\lambda(M) = \det(A-\lambda I)\det(D -\lambda I) - \det(B)\det(C) = \det(A-\lambda I)\det(D -\lambda I) - \det(BC) \\ =\det(A-\lambda I)\det(D -\lambda I)$


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