Say I have a matrix $A$ which is "almost" block diagonal. Meaning, the blocks on the diagonal might overlap by one element. For example:
$$A= \begin{bmatrix}a&b&c&0&0\\d&e&f&0&0\\g&h&i&j&0 \\ 0&0&k&l&m \\ 0&0&0&n&p \end{bmatrix}$$
Here, we have three blocks - one is 3-dimensional and two are 2-dimensional.
Is there anything interesting I can say about the eigenvalues/eigenvectors of $A$ in terms of eigenvalues/eigenvectors of the smaller "blocks"?
In the sense of linear maps, we can think of $A$ as representing a map which is a sum of linear maps acting on smaller subspaces, which might have one-dimensional intersection (but only between subspaces which correspond to adjacent "blocks"). I have good understanding of each "small" linear map (and also I know in general how the "intersection" elements should look like), and I would like to be able to say something about the eigenvalues of the entire linear map.
Clearly this is more complicated than just diagonalizing each individual block, but maybe since the intersection is very small, I can still say something? Maybe even give a rough estimate for the eigenvalues which depends on the norm of the linear map when restricted to the intersecting subspaces?
Thanks in advance.