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Matrix Analysis

Roger A. Horn and Charles R. Johnson
Cambridge University Press
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
William J. Satzer
, on

In this second edition of a well known text on matrix analysis, the authors retain the goals they had described in the preface to the first edition. They say, “… we present classical and recent results of matrix analysis that have proved to be important to applied mathematics. The book may be used as an undergraduate or graduate text and as a self-contained reference for a variety of audiences.” The authors say that only an elementary linear algebra course and knowledge of rudimentary analytical techniques are prerequisites.

The new edition retains most of the organizational form of the original book but adds a good deal of material. New work since 1985 is reflected in both a modified presentation of some topics and the inclusion of some new ones. One of these is the Jordan canonical form of a rank one perturbation; this was apparently driven by a question about updating the famous Google matrix. (Related questions arise in areas like signal processing where one would like to update matrix factorizations based on new data of rank one.) Another measure of the changes since the first edition are the number of exercises (now more than 1100 versus about 690 before), and the number of items in the index (now more than 3500 compared to 1200.) The latter change is particularly valuable to those who would use this book as a reference.

For readers unfamiliar with the first edition, the work “analysis” in the title could be misleading. There is, for example, no discussion of the calculus of matrices or of matrix functions like trace or determinant. Also, the matrix exponential is not discussed. Instead, this work stays much closer to the algebraic side. The major topics are eigenvalues, canonical forms, matrix factorizations, special matrices (Hermitian, positive definite or semidefinite, positive and negative), similarity, vector and matrix norms, and unitary equivalence. Many results from across the matrix analysis literature are available in this book, including many that are not readily accessible elsewhere. The text has a specialized feel to it and is best approached with a solid background in linear algebra.

The book would be a stretch for many undergraduates. It is dense and demands a fair amount of mathematical maturity and persistence. There are very few routine exercises and the hints provided in an appendix are rather sketchy.

Questions about numerical computation also don’t get any attention. That’s not a complaint – the book is already plenty long as it is. A good parallel text to fill in the computational aspects might be Golub and Van Loan’s Matrix Computations.

This is a solid reference and a pretty good deal at this price.

Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

1. Eigenvalues, eigenvectors, and similarity
2. Unitary similarity and unitary equivalence
3. Canonical forms for similarity, and triangular factorizations
4. Hermitian matrices, symmetric matrices, and congruences
5. Norms for vectors and matrices
6. Location and perturbation of eigenvalues
7. Positive definite and semi-definite matrices
8. Positive and nonnegative matrices
Appendix A. Complex numbers
Appendix B. Convex sets and functions
Appendix C. The fundamental theorem of algebra
Appendix D. Continuous dependence of the zeroes of a polynomial on its coefficients
Appendix E. Continuity, compactness, and Weierstrass's theorem
Appendix F. Canonical pairs.