| this thesis we study two different problems related to eigenvalue error bounds. In the first part of our thesis, we examine a conjecture of Knyazev and Argentati [Siam J. Matrix Anal. Appl., 29 (2006), pp. 15--32 ] bounding the difference between Ritz values of a Hermitian matrix A for two subspaces, one of which is A-invariant. We provide a proof for a slightly weaker version of the conjecture, and discuss the recently published full proof. Moreover we give implications of the now proven bound and examine how it compares to a classical bound in the same context. In the second part of our thesis, we derive some properties of complex Hessenberg matrices and consider the relevant normal matrix cases of these to re-examine the lengths of the Ritz vectors in the rounding error analysis of the Lanczos process for tridiagonalizing certain normal matrices. This question has already been studied for the real symmetric case, but part of that answer has never been published in scientific journals, and in that case we give new theory. For the more general normal matrix cases we develop applicable theory including some new tight bounds. |
