Variational principles and numerical algorithms for symmetric matrix pencils

Abstract

This thesis concerns the eigenvalue problems for symmetric ( or hermitian) matrix pencils lambda A-B in which A is nonsingular and neither A nor B is definite. Our intention is to find out to what extent some classical theoretical results and numerical algorithms for symmetric matrices can be carried over to symmetric pencils. First, a spectral characterization of definite pencils is presented, and an inertia function is introduced and used to give a simple algorithm for finding a positive definite matrix in a definite pencil. Then the minimax theorems are developed in Chapter 2 and its application to positive semidefinite perturbations is included. Following that, we study the numerical methods. The Rayleigh quotient iteration is introduced and the local and global convergence properties are established. Moreover, a method based on minimization of the Rayleigh quotients is proposed for definite pencils. Then the Rayleigh-Ritz method is formulated for the symmetric pencil problem, and using Krylov subspaces, an approximation error bound is proved. In particular, Lanczos' algorithm is discussed and a convergence criterion is demonstrated by using residuals or by a local perturbation expansion.