Perturbation of Partitioned Hermitian Generalized Eigenvalue Problem (original) (raw)

This paper is concerned with the Hermitian positive definite generalized eigenvalue problem A − λB for partitioned matrices A = A 11 A 22 , B = B 11 B 22 , where both A and B are Hermitian and B is positive definite. We present bounds on how its eigenvalues vary when A and B are perturbed by Hermitian matrices. 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 positive 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 in Stewart and Sun (Matrix Perturbation Theory, Academic Press, Boston, 1990, p.300.) that different co...