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.