Several Concepts to Investigate Strongly Nonnormal Eigenvalue Problems

Abstract

Eigenvalue analysis plays an important role in understanding physical phenomena. However, if one deals with strongly nonnormal matrices or operators, the eigenvalues alone may not tell the full story. A well-known tool which can be useful to get more insight in the reliability or sensitivity of eigenvalues is $\varepsilon$-pseudospectra. Apart from $\varepsilon$-pseudospectra, we consider other tools which might help to learn more about the eigenvalue problem, viz., condition numbers of the eigenvalues, condition numbers of sets of eigenvectors, and angles between invariant subspaces. All these concepts will be studied and compared for both standard and generalized eigenvalue problems. The tools can be used to analyze large eigenvalue problems. We apply the different concepts to a generalized eigenvalue problem obtained from magnetohydrodynamics. In this problem one is interested in an interior part of the spectrum, called the Alfven spectrum.