Several Concepts to Investigate Strongly Nonnormal Eigenvalue Problems

Eigenvalue analysis plays an important role in understanding physical phenomena. However, if one deals with strongly nonnormal matrices or operators, the eigen-values alone may not tell the full story. A popular tool which can be useful to get more insight in the reliability or sensitivity of eigenvalues is "-pseudospectra. Apart from "-pseudospectra we consider other tools which might help to learn more about the eigen-value 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 diierent concepts to a generalized eigenvalue problem obtained from magnetohydrodynamics. In this problem one is interested in an interior part of the spectrum, called the Alfv en spectrum.