The effect of the perturbation of hermitian matrices on their eigenvectors

We show that under some appropriate normalization, the eigenvectors corresponding to the maximal and minimal eigenvalues of a hermitian matrix subjected to a small perturbation by a positive semidefinite matrix decrease and increase in length, respectively. It is also shown that an eigenvector of a general matrix corresponding to an eigenvalue which increases in modulus must, if normalized in some particular fashion, eventually decrease in length if the matrix undergoes a sufficiently large perturbation.