Exclusion Theorems and the Perturbation Analysis of the Generalized Eigenvalue Problem

A Bauer–Fike type theorem is proved for the eigenvalue problem $Ax = \lambda Bx$. A generalization is then applied to obtain perturbation bounds for clusters of eigenvalues. The conditioning of an eigenvalue (finite or infinite) is proved to be dependent on a Jordan type condition number, its eigenvector deficiency and the regularity of the matrix pencil $(A - \lambda B)$. An example illustrating the perturbation bounds is given.