Eigenvalues of a block matrix composed of Toeplitz matrices

Question

If I have a block matrix of the form

$$ M = \begin{pmatrix} A &B \\[6pt] -B & C \end{pmatrix} $$ and if $A$ is invertible I can write determinant in terms of the Schur complement as $$ det(M) = det(A)det(C+B A^{-1} B) $$

and as such the characteristic equation is $$ det(M-\lambda I) = det(A-\lambda I)det(C+B(A-\lambda I)^{-1} B) = 0 $$ If all the blocks are Toeplitz matrices, and non-zero only on and around the primary diagonal, is there some way to get at the eigenvalues of $M$ in terms of the eigenvalues of $A$,$B$, and $C$ without resorting to this characteristic equation?

Essentially I would like to put upper bounds on the max eigenvalue of $M$. I could use the Gershgorin circle theorem directly on $M$, however given that, even though the block components of $M$ are non-zero only near their primary diagonal, this is of course not true of $M$ itself. As such I would expect the circle theorem to put rather poor bounds on it's eigenvalues. I was hoping to get around this by moving instead to look at the block components and applying the circle theorem to them instead, as they really will be populated only near their primary diagonals.

Any and all help/comments appreciated.

EDIT: let's assume, if it helps, that all the blocks commute (they will contain only diagonal matrices of constants and derivative operators which I will assume always commute though this depends on the finite difference scheme)

Answer 1

Introduce the projection operator $P$ such that $A=PMP$, $Q=1-P$, and $C=QMQ$. Next consider the self-energy operator $\Sigma_P(E)=PMQ(E-QMQ)^{-1}QMP$.The map $M\rightarrow PMP+\Sigma_P(E)$ relates the eigenvalue problem on the full space to that on its subspace.

$[M-\lambda I]V=0 \leftrightarrow [PMP+\Sigma_P(\lambda)-\lambda PIP]PV=0$ or equivalently you have to solve the nonlinear eigenvalue problem $[A-B(\lambda-C)^{-1}B-\lambda PIP]PV=0$