Localized Eigenvectors from Widely Spaced Matrix Modifications

We start with a large matrix A whose structure is simple, say, with unit entries on the first subdiagonal and superdiagonal. Its eigenvalues and eigenvectors are known. We modify A in M widely spaced rows and columns. Then the "new eigenvectors" are nearly a sum of spikes xj = t|j-r| centered at the positions r of the modified rows. The new eigenvalues are given almost exactly by $\pm \sqrt{4+\mu^2}$, where $\mu$ is an eigenvalue of the M by M modification. We extend this analysis to a larger class of structured matrices. For a banded Toeplitz matrix, our experiments show similar spikes centered around modified rows, and we have a conjecture on the structure of the new eigenvectors. For a single diagonal modification of the adjacency matrix of an infinite two-dimensional grid, we find the new eigenvalue from an elliptic integral (and we don't yet recognize the eigenvector).