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Matrix Theory: From Generalized Inverses to Jordan Form

Robert Piziak and P. L. Odell
Publisher: 
Chapamn & Hall/CRC
Publication Date: 
2007
Number of Pages: 
548
Format: 
Hardcover
Series: 
Pure and Applied Mathematics 288
Price: 
89.95
ISBN: 
9781584886259
Category: 
Monograph
[Reviewed by
Henry Ricardo
, on
01/11/2008
]

This book is intended for a second course in linear algebra at the advanced undergraduate or beginning graduate level. As the title indicates candidly, the book is (complex) matrix-oriented, and the only mention of linear transformations I can find is in Appendix D (“A Review of Basics”). The word “field” is never mentioned, nor does the term “module” appear. Those looking for a graduate level text with emphasis on abstraction and generality would be better advised to consider a text such as Roman’s Advanced Linear Algebra.

Once the premises are understood, the reader and potential adopter will find this to be a solid book written in a conversational and friendly style (the words “freebee” and “neat” occur often). Most chapters begin with a boxed list of terms to be introduced, such as

Hermitian inner product, parallelogram Law, polarization identity, Apollonius identity, Pythagorean theorem.

 

The Moore-Penrose inverse is developed early, leading to a full treatment of generalized inverses. The “big finish” to the course is the Jordan canonical form. This is followed by a final chapter on various “Multilinear Matters,” including a discussion of bilinear forms and the tensor product of matrices. By the authors’ own admission, the characterization of nilpotent matrices in Chapter 3 is “probably the most challenging material in the book,” but the exposition and proof are done nicely.

There are numerous concrete examples and an abundance of exercises. Every chapter has at least one “MATLAB moment” which provides commands and/or functions for various matrix calculations, factorizations, and so forth. Many sections end with suggestions for further reading, often articles in MAA journals. Historical tidbits are scattered throughout the book. There are four appendices covering basic background material.

Matrix Theory contains no true “applications” and little discussion of numerical issues except for some “numerical notes” on topics such as pivoting strategies and operation counts. These are useful, but not more informative than the remarks to be found in a good undergraduate introduction to linear algebra. There are similarities to The Theory of Matrices, by Lancaster and Tismenetsky, but the latter book clearly aims to prepare readers for certain applications and discusses variational methods, perturbation theory, and stability theory of systems of differential equations — all missing from Piziak-Odell. However, the book under review is more accessible than the text-cum-reference work of Lancaster and Tismenetsky. I would recommend the book under review to anyone seeking a text for a follow-up course in matrix theory.


Henry Ricardo (henry@mec.cuny.edu) is Professor of Mathematics at Medgar Evers College of The City University of New York and Secretary of the Metropolitan NY Section of the MAA. His book A Modern Introduction to Differential Equations was published by Houghton Mifflin in January, 2002; he is currently writing a linear algebra text.

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