The Denite Generalized Eigenvalue Problem: A New Perturbation Theory 1
Roy Mathias, Chi-Kwong Li · 2004
Let (A; B) be a denite pair of n n Hermitian matrices. That is, jx Axj + jx Bxj 6 0 for all non-zero vectors x 2 C n . It is possible to nd an n n non-singular matrix X with unit columns such that where j and j are real numbers. We call the pairs ( j; j) normalized generalized eigenvalues of the denite pair (A; B). These pairs have not been studied previously. We rework the perturbation theory for the eigenvalues and eigenvectors of the denite generalized eigenvalue problem Ax = Bx 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) = minfjx (A + iB)xj : x 2 C n ; x x = 1g