  We derive new perturbation bounds for eigenvalues of Hermitian matrices with
block structures. The structures we consider range from a standard 2-by-2 block
form to block tridiagonal and tridigaonal forms. The main idea is the
observation that an eigenvalue is insensitive to componentwise perturbations if
the corresponding eigenvector components are small. We show that the same idea
can be used to explain two well-known phenomena, one concerning extremal
eigenvalues of Wilkinson's matrices and another concerning the efficiency of
aggressive early deflation applied to the symmetric tridiagonal QR algorithm.
