The back-cover of the brand-new second edition of Denis Serre’s Matrices states that “[w]ith forty percent new material, this second edition is significantly different from the first” and we quickly get an enumeration of seven new topics including, e.g., “tensor and exterior calculus [and] polynomial identities,” “Weyl’s and von Neumann’s inequalities,” and “[the] Jacobi method with random choice.” It’s obvious from the start, therefore — indeed, even before opening the book — that there is a lot of rather exotic (and exciting) material bound in these pages, well beyond the usual scope of a course on matrices. But then again, courses on matrices as such, as opposed to courses in linear algebra, are exotic things in themselves.
We tend to think of this material as undergraduate fare, but Serre’s book is aimed at a graduate student audience, so there must be something unusual going on. But what? Well, it is of course instructive to let him speak for himself:
In some sense this is not a specialized book. For instance, it is not as detailed as [other mainstream books] concerning numerics, or … eigenvalue problems, or … Weyl-type inequalities. But it covers, at a slightly higher than basic level, all these aspects, and is therefore well suited for a graduate program.
Thus, Matrices is structured so as to cut across a number of boundaries and collect together a number of important and often rather advanced applications of, and variations on, the theory of matrices in one place: an interesting and worthwhile endeavor, obviously, and clearly vouchsafed only to a more mature audience.
Regarding the undergraduate level linear algebra that a reader of Matrices must have under his belt, Serre notes that, regarding his first edition, he
felt ashamed of the very light presentations of the backgrounds in linear algebra and elementary matrix theory in [his erstwhile] Chapter 1 … I thus began by rewriting this part completely, taking the opportunity to split pure linear algebra from the introduction to matrices. I hope that the reader is satisfied with the new Chapters 1 and 2…
And so it is that Serre’s first two chapters present a terse but adequate review of the indicated undergraduate basics, from “vectors and scalars” to “matrices and bilinear forms” sans the more “technical” material regarding matrices, i.e. the springboard to what he is up to in what follows, which appears in his third chapter, “square matrices” where much of the emphasis is placed on the eigenvalue problem, characteristic polynomials, and diagonalization and triagonalization.
Graduate level material is hit with the (short) fourth chapter, “tensor and exterior products,” which is obviously an exposition of basic multilinear algebra, after which Serre proceeds to considerably more specialized themes, some of which have already been mentioned. Suffice it to say by way of further illustration that, e.g., surrounding von Neumann’s inequality, some functional analysis (norms of continuous operators, etc.) is featured; more functional analysis (norm theory, again, really) is needed vis à vis stochastic matrices (in Chapter 8), and Chapter 10 deals with material flirting with Lie theory (its title being “Exponential of a matrix, decomposition, and classical groups”). The last three chapters are apparently more exotic still — possibly by far; the last chapter, for example, is devoted to the important theme of approximating eigenvalues.
The prose is good, there are plenty of exercises, and the book is accordingly well suited for self-study. Additionally Matrices would serve very well as a text for courses locally homeomorphic to Serre’s, of which there should clearly be more.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.