Book Review: Matrix Analysis and Applied Linear Algebra, 2nd Edition

When I discovered the first edition of Carl D. Meyer’s book on “Matrix Analysis and Applied Linear Algebra”, I considered it the best book on Linear Algebra I was aware of — and surely the best available book for an application-minded reader. In short: the book I really wished had been available when I was a graduate student and encountering this material for the first time.

So it was with great excitement and curiosity when I discovered that a new, second edition was available. Would it be possible to make a near-perfect book even better?

The first thing to notice is that this new edition is not a second, revised edition of the previous one — it is an entirely new book. That’s already a little strange: if the previous version was so good, why undertake a complete rewrite?

Upon closer inspection, it turns out that this current edition is not even a rewrite of the previous one, but a completely different effort altogether: a new book, with a new focus, and for a different audience. That starts to be outright disturbing.

The new edition does not attempt to be a conventional textbook on conventional linear algebra. The treatment of the classical topics in an introductory (graduate-level) course on linear algebra (vector spaces, basis, linear transformation, etc) seem to receive less care and attention. Instead, there is an emphasis on and preparation for work in numerical computations: we find frequent excursions into numerical applications and matrix calculations — and specifically large-scale matrix computations. That’s all good and well, but the implementation of numerical linear algebra routines is still a game for specialists, while other fields (theoretical physics comes to mind) use linear algebra concepts daily, even if they never calculate numerical values for matrix entries explicitly.

An explanation for this approach, and a bit of a philosophy of the entire second edition, is provided roughly in the middle of the work, in a section entitled “Historical (and Editorial) Comments” on p. 522f. Here, the author takes a very strong stance against the abstract, coordinate-free, determinant-free, “algebraic” approach to linear algebra, and instead opines that the arrival of computers did “change everything” (author’s emphasis), and therefore necessitates a different approach to teaching linear algebra, one that an emphasizes computational algorithms and their challenges, and avoids what he refers to as the customary “sterilize(d) presentation of linear algebra”.

Well, maybe. The computer revolution opened up vast new fields of new applications. But it did not change, for instance, the tremendous power of learning to think in terms of abstract spaces and mappings — unencumbered by any regard for how that mapping might be implemented (or even defined)! The arrival of computers also did not change the foundations of classical or quantum mechanics, both of which rely on all the basic notions and techniques of linear algebra.

Overall, this section made a strange impression on me — unnecessarily belligerent, iconoclastic, almost feverish. It also appears hastily written, with some of the arguments not carried through carefully enough, and some of the wording needlessly partisan. (The impression of haste and lack of final polish in arguments also struck me more generally when looking through this book.)

But it does explain what this book attempts to do. (By the way, this is the kind of discussion that the reader would expect in the preface; why it is buried near the geometric center of the work is anyone’s guess.) The author wants to prepare students for “Applied Linear Algebra for the 21st Century”, so to speak. That’s not just a perfectly valid goal, but a truly commendable one: the danger is very real that the teaching of canonical topics goes stale.

But there are (at least) two problems with that: the first one is that (in my opinion) the presentation of linear algebra, as provided by the author himself in the first edition, has by no means gone stale! The second problem is that the author does not really succeed in carrying out his program: the resulting presentation feels over-ambitious, uneven, and unfinished — arguably the worst a textbook can be.

For instance, we find a chapter on “Inner Product Spaces and Fourier Expansion”, devoting 70 pages (!) to Fourier expansions and the Fast Fourier Transform. No question: important topics. But is an introductory book on linear algebra really the right place and context for them? The material is certainly interesting, for example when the author presents the Singular Value Decomposition as a (formal) Fourier expansion. The problem is that Fourier expansion properly takes place in an infinite-dimensional space of integrable functions, whereas the presentation here operates in a finite-dimensional vector space of singular vectors. Will students be helped or confused by this sleight-of-hand? The presentation then jumps to Latent-Semantic Indexing, Principal Component Analysis, and Cluster Analysis — all fascinating topics, but complicated enough to deserve their own treatment, once linear algebra has been mastered: which is precisely why I believe that understanding linear algebra concepts is so important.

Overall, this new edition feels more like an overloaded, uneven, and at times almost haphazard collection of fascinating, though often quite advanced topics. (The last chapter, for example, introduces Tosio Kato’s resolvent calculus and perturbation theory for eigenvalues — not a topic usually covered in an introductory text.) Like its predecessor this book is chock-full of fascinating facts and observations (e.g. the fact that the inverse $A^{-1}$ of a matrix is a polynomial in $A$ (p. 313), or the consequences of Sylvester’s Law of Inertia for the signs of eigenvalues (p. 399)).

Whereas the first edition served brilliantly as well-rounded, comprehensive, and careful introduction to linear algebra at the graduate level, the new edition seems to be more a compendium of topics for experienced practitioners. Readers with the appropriate background and knowledge will find the sheer range of topics, and their often interesting and unexpected aspects, fascinating, valuable, and stimulating. A beginner, not so much.