Eigenvalue perturbation bounds for Hermitian block tridiagonal matrices

We derive new perturbation bounds for eigenvalues of Hermitian matrices with block tridiagonal structure. The main message of this paper is that an eigenvalue is insensitive to blockwise perturbation, if it is well-separated from the spectrum of the diagonal blocks nearby the perturbed blocks. Our bound is particularly effective when the matrix is block-diagonally dominant and graded. Our approach is to obtain eigenvalue bounds via bounding eigenvector components, which is based on the observation that an eigenvalue is insensitive to componentwise perturbation if the corresponding eigenvector components are small. We use the same idea to explain two well-known phenomena, one concerning aggressive early deflation used in the symmetric tridiagonal QR algorithm and the other concerning the extremal eigenvalues of Wilkinson matrices.