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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]

Suppose A-BXB*, A-BX -X*B*, and A-BX +X*B* are three linear matrix expressions over the field of complex numbers, where A is an Hermitian or skew-Hermitian matrix. In this paper, we consider how to choose an Hermitian or skew-Hermitian matrix X such that A-BXB* have the maximal and minimal possible ranks, and how to choose X such that A - BX ± X*B* attain the minimal possible ranks. Some applications to Hermitian or skew-Hermitian solutions of matrix equations with symmetric patterns are also given. [ABSTRACT FROM AUTHOR]