The theory of matrices is, of course. a lot richer than what we are able to foist on our students in undergraduate school. This is the case even if we get them for a full two semesters, so that the second course is something like “advanced linear algebra” and we opt to go computational. Actually I generally don’t: I use my second semester for what in the heroic days of my youth used to be part of the first semester, when linear algebra meant Halmos (or something kindred — whatever that means).
This is not to say that one can ever see enough good stuff about matrices: they are truly wonderful things. It used to be the case (again, many moons ago) that the place to go for more was Gantmacher’s book(s: there are two volumes in the Chelsea version), and before you knew it, you’d be hip deep in applications of various types, interesting to be sure, but nowadays largely relegated to other courses in algebra. And of course this is the case with so much of this material — we quickly tend to get more abstract these days, for better or for worse. What we’re dealing with in the book under review is happily rather different.
Before we get to specifics it is important to note that what Zhan is up to in his Matrix Theory is also rather different in flavor from what Denis Serre does in his book, Matrices: Theory and Applications, which I reviewed in this column not all that long ago. Briefly put, Zhan is far more heavily focused on matrices as Dingen in sich, while Serre is more oriented toward multilinear algebra and (e.g.) Lie theory. In point of fact, Zhan doesn’t even allow Lie into his Index
All right, then, what does Zhan do? Well, in an orbit of some 250 pages or so he travels from where a good undergraduate course (even in today’s model) leaves off, i.e. diagonalizability and the eigenvalue problem, some spectral stuff, and the business of matrix decompositions — oops, we’re already beyond what most undergraduate courses do — and Gröbner bases, and then hits a host of rather marvelous themes including the inner life of Hermitian matrices and matrix perturbation theory, as well as some pretty exotic material such as the Frobenius-König Theorem and Perron-Frobenius theory. The latter has to do with positive matrices and their eigenvalues and eigenvectors and Zhan follows Wielandt’s approach — the same Wielandt, of course, who gave such an appealing proof of the Sylow Theorems. There is also material on completion of partial matrices, which he motivates by nothing less than the Netflix problem (see p. 149): we’re beaming back down to earth.
The book’s last chapters are devoted to miscellaneous topics and connections with combinatorics, number theory, algebra, geometry, and polynomials, and the book is capped off by a compact section on unsolved problems. A propos, regarding number theory Zhan features Hilbert’s Nulstellensatz and employs Noether’s normalization lemma (and a Sylvester matrix) in its proof (due to Arrondo), while the first unsolved problem on his list (of twenty) is the conjecture that for every n (a positive integer, of course) there’s a Hadamard matrix of order 4n; recall that a Hadamard matrix is a square matrix of 1’s and –1’s with rows (and columns) mutually orthogonal.
There are plenty of exercises to be had, and the author’s goal is clearly to guide able and willing graduate students toward research in this area, which certainly possesses the attractive qualities of being both accessible (you don’t need to read Grothendieck’s SGA to get started on research) and exciting — it’s algebra after all! I think Zhan will be successful in this enterprise: it’s a very nice book indeed.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.