On perturbations of matrix pencils with real spectra

Abstract

Perturbation bounds for the generalized eigenvalue problem of a diagonalizable matrix pencil A − λ B A - \lambda B with real spectrum are developed. It is shown how the chordal distances between the generalized eigenvalues and the angular distances between the generalized eigenspaces can be bounded in terms of the angular distances between the matrices. The applications of these bounds to the spectral variations of definite pencils are conducted in such a way that extra attention is paid to their peculiarities so as to derive more sophisticated perturbation bounds. Our results for generalized eigenvalues are counterparts of some celebrated theorems for the spectral variations of Hermitian matrices such as the Weyl-Lidskii theorem and the Hoffman-Wielandt theorem; and those for generalized eigenspaces are counterparts of the celebrated Davis-Kahan sin ⁡ θ , sin ⁡ 2 θ \sin \theta ,\sin 2\theta theorems for the eigenspace variations of Hermitian matrices. The paper consists of two parts. Part I is for generalized eigenvalue perturbations, while Part II deals with generalized eigenspace perturbations.