Extremal rays in the Hermitian eigenvalue problem

Perturbation of Partitioned Hermitian Generalized Eigenvalue Problem

We are concerned with the perturbation of a multiple eigenvalue μ of the Hermitian matrix A = diag(μI,A22) when it undergoes an off-diagonal perturbation E whose columns have widely varying magnitudes. When some of E’s columns are much smaller than the others, some copies of μ are much less sensitive than any existing bound suggests. We explain this phenomenon by establishing individual perturb...

Quadratic Residual Bounds for the Hermitian Eigenvalue Problem

Let A = " M R R N # and ~ A = " M 0 0 N # be Hermitian matrices. Stronger and more general O(kRk 2) bounds relating the eigen-values of A and ~ A are proved using a Schur complement technique. These results extend to singular values and to eigenvalues of non-Hermitian matrices. (1) be Hermitian matrices. Since jjA ? ~ Ajj = jjRjj one can bound the diierence between their eigenvalues in terms of...

An extremal eigenvalue problem arising in heat conduction

This article is devoted to the study of two extremal problems arising naturally in heat conduction processes. We look for optimal configurations of thermal axisymmetric fins and model this problem as the issue of (i) minimizing (for the worst shape) or (ii) maximizing (for the best shape) the first eigenvalue of a selfadjoint operator having a compact inverse. We impose a pointwise lower bound ...

Changing poles in the rational Lanczos method for the Hermitian eigenvalue problem

Applications such as the modal analysis of structures and acoustic cavities require a number of eigenvalues and eigenvectors of large scale Hermitian eigenvalue problems. The most popular method is probably the spectral transformation Lanczos method. An important disadvantage of this method is that a change of pole requires a complete restart. In this paper, we investigate the use of the ration...

EXTREMAL EIGENVALUE PROBLEMS FOR THE LAPLACIANSteven