Considering a collection of numbers (or other mathematical objects) as a rectangular array is an idea used by many people from various branches of mathematics, statistics, computer sciences, and engineering. The properties of such arrays, which we call matrices, are also at the base of recent developments in combinatorics, coding theory, and many other subjects.
As its title indicates, the book under review is a concise account of the theory of matrices, giving a quick but powerful picture of the theory focusing mostly on inequalities. The book has three main parts. The first part gives a survey of matrix theory. It contains various lists of results, many of them with no proof. As a foundation for the next two parts, this survey includes some sections introducing numbers related to matrices, such as the determinant, permanent, trace, and characteristic roots (aka eigenvalues).
The second part, which is indeed the main part and heart of the book, contains various inequalities involving the above numbers for various kinds of matrices. One may find a list of them in the table of contents. Convexity plays an important role: it is the first topic discussed, and the two first sections study it. In the process, the authors give a quick review of important classical inequalities, which seems very nice for the people working on inequality theory. The third part of the book studies the location of characteristic roots by giving several inequalities involving them.
The book includes compactly presented information on many classical theorems, relations, and inequalities for matrices. This makes the book a friendly reference for researchers who deal with matrices, especially those who work with inequalities.
The book has no exercises, so is it not a textbook. But the structure of the chapters and the method of presentation are much like what one would find in a text. I believe that if an instructor considers some of the facts that are merely listed as exercises, then the book could be used as a textbook in an advanced course for undergraduates, or even as a part of some graduate courses.
I highly recommend this book for anyone involved in a course related to or using matrix theory: there is no doubt that they will find some very nice and useful information.
Mehdi Hassani is a faculty member at the Department of Mathematics, Zanjan University, Iran. His field of interest is Elementary, Analytic and Probabilistic Number Theory.