Matrix perturbations: bounding and computing eigenvalues

Despite the somewhat negative connotation of the word, not every perturbation is a bad perturbation. In fact, while disturbing the matrix entries, many perturbations still preserve useful properties such as the orthonormality of the basis of eigenvectors or the Hermicity of the original matrix. In the first part of this thesis, some of these property preserving perturbations are analyzed, with a focus on the consequences to the eigenvalues. For Hermitian rank-k perturbations of Hermitian matrices, this resulted in improved Weyl-type bounds for the perturbed eigenvalues. With respect to normal matrices, normality preserving normal perturbations were also considered and, for 2 x 2 and for rank-one matrices, the analysis is now complete. For higher rank, all essentially Hermitian normality perturbations are described. Moreover, the normality preserving augmentation of normal matrices is revisited and complemented with an analysis of the consequences to the eigenvalues. All augmentations resulting in normal matrices with eigenvalues on a quadratic curve in the complex plane are also constructed. In the second part, the Subspace Projected Approximate Matrix (SPAM) method, an iterative method for the Hermitian eigenvalue problem, is investigated. For certain choices of the approximation matrix, SPAM turns out to be mathematically equivalent to the Lanczos method. While more sophisticated approximations turn SPAM into a boosted version of the Lanczos method, it can also be interpreted as an attempt to enhance a certain instance of the preconditioned Jacobi-Davidson method.