Computing invariant subspaces of a general matrix when the eigensystem is poorly conditioned

Author:
J. M. Varah

Journal:
Math. Comp. **24** (1970), 137-149

MSC:
Primary 65.40

DOI:
https://doi.org/10.1090/S0025-5718-1970-0264843-9

MathSciNet review:
0264843

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Abstract | References | Similar Articles | Additional Information

Abstract: The problem of calculating the eigensystem of a general complex matrix is well known. In many cases, however, the eigensystem is poorly determined numerically in the sense that small changes in the matrix can cause large changes in the eigensystem. For these matrices, a decomposition into higher-dimensional invariant subspaces is desirable.

In this paper we define a class of matrices where this is true, and propose a technique for calculating bases for these invariant subspaces. We show that for this class the technique provides basis vectors which are accurate and span the subspaces well.

**[1]**F. L. Bauer, "Optimally scaled matrices,"*Numer. Math.*, v. 5, 1963, pp. 73-87. MR**0159412 (28:2629)****[2]**Chandler Davis & W. M. Kahan,*The Rotation of Eigenvectors by a Perturbation*. III, University of Toronto Computer Science Dept. Tech. Report, no. 6, 1968.**[3]**A. S. Householder,*Principles of Numerical Analysis*, McGraw-Hill, New York, 1953. MR**15**, 470. MR**0059056 (15:470b)****[4]**W. M. Kahan,*Inclusion Theorems for Clusters of Eigenvalues of Hermitian Matrices*, University of Toronto Institute of Computer Science Tech. Report, 1967.**[5]**Tosio Kato,*Perturbation Theory for Linear Operators*, Springer-Verlag, Berlin, 1966. MR**34**#3324. MR**0203473 (34:3324)****[6]**J. M. Varah, "The calculation of the eigenvectors of a general complex matrix by inverse iteration,"*Math. Comp.*, v. 22, 1968, pp. 785-791. MR**0240968 (39:2313)****[7]**J. M. Varah, "Rigorous machine bounds for the eigensystem of a general complex matrix,"*Math. Comp.*, v. 22, 1968, pp. 793-801. MR**0243731 (39:5052)****[8]**J. M. Varah,*The Computation of Bounds for the Invariant Subspaces of a General Matrix Operator*, Stanford University Computer Science Dept. Tech. Report, CS66, 1967.**[9]**J. H. Wilkinson,*The Algebraic Eigenvalue Problem*, Clarendon Press, Oxford, 1965. MR**32**#1894. MR**0184422 (32:1894)**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1970-0264843-9

Keywords:
Invariant subspaces,
eigenvectors,
ill-conditioned eigenvalue problem,
computation of eigensystems

Article copyright:
© Copyright 1970
American Mathematical Society