The Definite Generalized Eigenvalue Problem: A New Perturbation Theory

Let (A, B) be a definite pair of n × n Hermitian matrices. That is, |x∗Ax| + |x∗Bx| 6= 0 for all non-zero vectors x ∈ C. It is possible to find an n × n non-singular matrix X with unit columns such that X∗(A + iB)X = diag(α1 + iβ1, . . . , αn + iβn) where αj and βj are real numbers. We call the pairs (αj, βj) normalized generalized eigenvalues of the definite pair (A, B). These pairs have not been studied previously. We rework the perturbation theory for the eigenvalues and eigenvectors of the definite generalized eigenvalue problem βAx = αBx in terms of these normalized generalized eigenvalues and show that they play a crucial rule in obtaining the best possible perturbation bounds. In particular, in existing perturbation bounds, one can replace most instances of the Crawford number c(A, B) = min{|x∗(A + iB)x| : x ∈ C, x∗x = 1} with the larger quantity dmin = min{|αj + iβj| : j = 1, . . . , n}. This results in bounds that can be stronger by an arbitrarily large factor. We also give a new measure of the separation of the jth eigenvalue from the kth: |(αj + iβj) sin(arg(αj + iβj) − arg(αk + iβk))|. This asymmetric measure is entirely new, and again results in bounds that can be arbitrarily stronger than the existing bounds. We show that all but one of our bounds are attainable. We also show that the Crawford number is the infimum of the distance from a definite pencil, a fortiori diagonalizable, to a non-diagonalizable pair. AMS(MOS) 65F15, 65F35, 15A18, 15A60