On the Sensitivity of the Eigenvalue Problem $Ax = \lambda Bx$

Abstract

This paper considers the sensitivity of the eigenvalues and eigenvectors of the generalized matrix eigenvalue problem $Ax = \lambda Bx$ to perturbations of A and B. The notion of a deflating subspace for the problem is introduced, and error bounds for approximate deflating subspaces obtained. The bounds also provide information about the eigenvalues of the problem. The resulting perturbation bounds estimate realistically the sensitivity of the eigenvalues, even when B is singular or nearly singular. The results are applied to the important special case where A is Hermitian and B is positive definite.