Matrices. Theory and applications. Transl. from the French. 2nd ed. (English) Zbl 1206.15001
Graduate Texts in Mathematics 216. New York, NY: Springer (ISBN 978-1-4419-7682-6/hbk; 978-1-4419-7683-3/ebook). xiv, 289 p. (2010).
The first edition of this book (2002) was reviewed in Zbl 1011.15001, and the general comments there apply to the new edition.
However, the book has been considerably revised and the new edition is nearly 40% longer. The initial review of basic material in the first three chapters has been rewritten and strengthened. This is followed by a new chapter on tensor products of vector spaces and linear transformations which culminates with a simple proof due to S. Rosset [Isr. J. Math. 23, 187–188 (1976; Zbl 0322.15020)] of the Amitsur-Levitski theorem on the standard polynomial identity on \(M_{n}(K)\).
Other topics which have been added include: calculating with rank-1 perturbations; regularity of simple eigenvalues for complex matrices; functional analysis and the Dunford-Taylor analogue of the Cauchy integral formula for matrices; numerical range; the Weyl inequalities for eigenvalues of sums of Hermitian matrices; von Neumann’s inequality for linear contraction mappings; convergence of a version of the Jacobi method for computing eigenvalues. The author has added many new exercises extending the basic material and these add considerably to the value of the book.
Rather than a systematic exposition of one part of the subject, the author touches on many aspects (algebra, functional analysis and computation), offering a range of topics, and presenting some fascinating and perhaps not familiar parts of the subject. Even those familiar with the subject may find something new here, and a good student, willing to work hard, could find it a rewarding book for self-study.
However, the book has been considerably revised and the new edition is nearly 40% longer. The initial review of basic material in the first three chapters has been rewritten and strengthened. This is followed by a new chapter on tensor products of vector spaces and linear transformations which culminates with a simple proof due to S. Rosset [Isr. J. Math. 23, 187–188 (1976; Zbl 0322.15020)] of the Amitsur-Levitski theorem on the standard polynomial identity on \(M_{n}(K)\).
Other topics which have been added include: calculating with rank-1 perturbations; regularity of simple eigenvalues for complex matrices; functional analysis and the Dunford-Taylor analogue of the Cauchy integral formula for matrices; numerical range; the Weyl inequalities for eigenvalues of sums of Hermitian matrices; von Neumann’s inequality for linear contraction mappings; convergence of a version of the Jacobi method for computing eigenvalues. The author has added many new exercises extending the basic material and these add considerably to the value of the book.
Rather than a systematic exposition of one part of the subject, the author touches on many aspects (algebra, functional analysis and computation), offering a range of topics, and presenting some fascinating and perhaps not familiar parts of the subject. Even those familiar with the subject may find something new here, and a good student, willing to work hard, could find it a rewarding book for self-study.
Reviewer: John D. Dixon (Ottawa)
MSC:
| 15-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra |
| 15A60 | Norms of matrices, numerical range, applications of functional analysis to matrix theory |
| 15A42 | Inequalities involving eigenvalues and eigenvectors |
| 15B48 | Positive matrices and their generalizations; cones of matrices |
| 65F15 | Numerical computation of eigenvalues and eigenvectors of matrices |
| 65F10 | Iterative numerical methods for linear systems |
| 20G20 | Linear algebraic groups over the reals, the complexes, the quaternions |
| 15A23 | Factorization of matrices |
| 15A04 | Linear transformations, semilinear transformations |
