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**Matrix perturbation theory.**
*(English)*
Zbl 0706.65013

Computer Science and Scientific Computing. Boston etc.: Academic Press, Inc. xv, 365 p. $ 49.95 (1990).

Matrix perturbation theory, chosen and described in a way appropriate for numerical analysis, is developed starting with chapters on QR decomposition and projections, eigenvalues with the Schur decomposition, the Jordan canonical form and the field of values, and followed by chapters on the singular value decomposition and CS decomposition. A wealth of results on matrix norms, especially the unitarily invariant ones, linear systems and pseudoinverses and perturbation of eigenvalues are given. The concluding chapters deal with invariant subspaces and generalized eigenvalue problems, and here as well as in the earlier chapters, many achievements of the last quarter century, during which time the authors of the monograph under review have been among the most active contributors, are neatly integrated with that standard knowledge where we hitherto have used to refer to the books by A. S. Householder [The theory of matrices in numerical analysis. New York etc.: Blaisdell Publishing Company (1964; Zbl 0161.12101)] or I. C. Gohberg and M. G. Krein [Introduction to the theory of linear nonselfadjoint operators. Providence, RI: American Mathematical Society (1965; Zbl 0181.13504)] or the paper by C. Davis and W. M. Kahan [SIAM J. Numer. Anal. 7, 1–46 (1970; Zbl 0198.47201)].

The authors have been careful in developing all theorems from first principles, and proofs are simple. Some misprints like those on pp. 178 (Ex. 2) and 274 (Ex. 1.5) just serve to keep the reader awake, while even this reviewer had some difficulty in finding out how it was intended to prove the convexity of the field of values on p. 23 and the definition on bottom of p. 193.

Anybody needing to analyze how a numerical algorithm responds to perturbations will now be able to lean on the results described in this monograph.

The authors have been careful in developing all theorems from first principles, and proofs are simple. Some misprints like those on pp. 178 (Ex. 2) and 274 (Ex. 1.5) just serve to keep the reader awake, while even this reviewer had some difficulty in finding out how it was intended to prove the convexity of the field of values on p. 23 and the definition on bottom of p. 193.

Anybody needing to analyze how a numerical algorithm responds to perturbations will now be able to lean on the results described in this monograph.

Reviewer: Axel Ruhe (Stockholm)

### MSC:

65Fxx | Numerical linear algebra |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

15-02 | Research exposition (monographs, survey articles) pertaining to linear algebra |

15A18 | Eigenvalues, singular values, and eigenvectors |

15A42 | Inequalities involving eigenvalues and eigenvectors |

15A60 | Norms of matrices, numerical range, applications of functional analysis to matrix theory |

15A09 | Theory of matrix inversion and generalized inverses |