Problems and results in matrix perturbation theory

Abstract

The perturbation theory is important in applications and theoretical investigations as well. Here we investigate three groups of perturbation problems which are related to computational methods of importance. The first section is related to the solution of linear systems of equations and a posteriori error estimates of the computed solution. The second section gives optimal bounds for the perturbations of LU factorizations. The final section gives a sharp upper bound for the eigenvalue perturbation of general matrices, which is better than the classical result of Ostrowski. We also show two applications of this result. The first application gives a sharp perturbation bound for the zeros of polynomials. The second application is related to a result of Edelman and Murakami on the backward stability of companion matrix type polynomial solvers. © 2009 Springer-Verlag Berlin Heidelberg.