This volume describes in monograph form important applications in numerical methods of linear algebra. The author presents material and extended results from recent papers in a readable style. The main goal of the book is to study the behaviour of the resolvent of a matrix under the perturbation by low rank matrices. Whereas the eigenvalues (the poles of the resolvent) and the pseudospectra (the sets where the resolvent takes large values) can move dramatically under such perturbations, the growth of the resolvent as a matrix-valued meromorphic function remains essentially unchanged. This has...