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We derive some properties of complex Hessenberg matrices and use the relevant normal matrix cases to examine the lengths of Ritz vectors in the rounding error analysis of the Lanczos tridiagonalization process. This question is important for the computational use of the process and has already been studied for the real symmetric matrix case, but because of its intricate and unedifying nature, part of the theory was never submitted to scientific journals. We develop a new and more palatable theory which also applies to Lanczos processes adapted to any form of normal matrix with collinear eigenvalues such as a Hermitian or skew-Hermitian matrix. The nonnormal matrix properties are intended to help in the analysis of the unsymmetric Lanczos process. [ABSTRACT FROM AUTHOR]

We present eigenvalue bounds for perturbations of Hermitian matrices and express the change in eigenvalues in terms of a projection of the perturbation onto a particular eigenspace, rather than in terms of the full perturbation. The perturbations we consider are Hermitian of rank one, and Hermitian or non-Hermitian with norm smaller than the spectral gap of a specific eigenvalue. Applications include principal component analysis under a spiked covariance model, and pseudo-arclength continuation methods for the solution of nonlinear systems. [ABSTRACT FROM AUTHOR]