PERTURBATION BOUNDS FOR THE DEFINITE GENERALIZED EIGENVALUE PROBLEM

Let A and B be Hermitian matrices, and let c(A, B) = inf{|xH(A + iB)x|:{norm of matrix} = 1}. The eigenvalue problem Ax = λBx is called definite if c(A, B)>0. It is shown that a definite problem has a complete system of eigenvectors and that its eigenvalues are real. Under pertubations of A and B, the eigenvalues behave like the eigenvalues of a Hermitian matrix in the sense that there is a 1-1 pairing of the eigenvalues with the perturbed eigenvalues and a uniform bound for their differences (in this case in the chordal metric). Pertubation bounds are also developed for eigenvectors and eigenspaces. © 1979.