7. Non-Hermitian Eigenvalue Problems
T. Chen, J. Demmel, Ming Gu, Yousef El-Mabruk Saad, Richard B. Lehoucq, D. C. Sorensen, Kristyn Maschhoff, Zhaojun Bai, David Day, Robert M. Freund, G.L.G. Sleijpen, Henk A. van der Vorst, Ran Li · Society for Industrial and Applied Mathematics eBooks · 2000
7.1 Introduction In this chapter we discuss the non-Hermitian eigenvalue problem (NHEP) Ax=λx, 7.1 where the square matrix . is called a right eigenvector. A vector satisfying y* A=λ y* 7.2 is called a left eigenvector of A. For a presentation of relevant theory for NHEPs, as well as pointers to literature, we refer to §2.5. In particular, we mention that the perturbation theory for these problems is delicate. One should keep in mind that a small norm of the residual for a computed eigenpair does not necessarily imply small errors in the computed values. For more information, see §2.5 and §7.13. Here we will give a brief introduction to those aspects that play a role in the selection of the algorithms. This will be followed by short characterizations of the different classes of methods, covered in this chapter. In contrast to a Hermitian matrix, a non-Hermitian matrix does not have an orthogonal set of eigenvectors; in other words, a non-Hermitian matrix A can in general not be transformed by an orthogonal matrix Q to diagonal form . Most non-Hermitian matrices can be transformed by a nonorthogonal X to diagonal form , but there exist matrices for which even this is not possible. Such matrices can be viewed as limit cases for which X converges to a singular operator, and these matrices do not have a complete set of eigenvectors; they are called defective. This reveals a source for numerical instability: if X is in some sense close to a singular operator, then the transformation may be expected to be sensitive to perturbations in A. Such perturbations are introduced by the process for the computation of X and D. Therefore, it is of great importance to work with orthogonal, or close to orthogonal, transformations as often as possible. It is well known that for any non-Hermitian matrix A there exists a unitary matrix U that transforms it to upper triangular form T= U* AU. 7.3