Perturbation of Partitioned Hermitian Definite Generalized Eigenvalue Problems

This paper is concerned with the Hermitian definite generalized eigenvalue problem A− λB for block diagonal matrices A 1⁄4 diagðA11; A22Þ and B 1⁄4 diagðB11; B22Þ. Both A and B are Hermitian, and B is positive definite. Bounds on how its eigenvalues vary when A and B are perturbed by Hermitian matrices are established. These bounds are generally of linear order with respect to the perturbations in the diagonal blocks and of quadratic order with respect to the perturbations in the off-diagonal blocks. The results for the case of no perturbations in the diagonal blocks can be used to bound the changes of eigenvalues of a Hermitian definite generalized eigenvalue problem after its off-diagonal blocks are dropped, a situation that occurs frequently in eigenvalue computations. The presented results extend those of Li and Li [Linear Algebra Appl., 395 (2005), pp. 183–190]. It was noted by Stewart and Sun [Matrix Perturbation Theory, Academic Press, Boston, 1990] that different copies of a multiple eigenvalue may exhibit quite different sensitivities towards perturbations. We establish bounds to reflect that feature, too. We also derive quadratic eigenvalue bounds for diagonalizable non-Hermitian pencils subject to off-diagonal perturbations.