If there is one thing that most of us can agree on, it’s that linear algebra is absolutely ubiquitous in mathematics and other areas, and, therefore, any properly prepared undergraduate mathematics major should have a firm background in it. Towards this end, many undergraduate mathematics departments offer two courses in the subject, one introductory and the other more advanced. At Iowa State University, for example, we offer the usual majors course and also a second course, dual-listed as carrying both undergraduate and graduate credit, titled Applied Linear Algebra and covering further topics.

While there is a general acknowledgment that a first course should cover the mainstays of the subject (matrices and determinants, vector spaces, linear transformations, inner product spaces and an introduction to eigen-things), there does not seem to be any firm consensus on what should be taught in a second course. Reflecting this, a number of books on advanced linear algebra, all with their own distinct personality and many using the phrase “advanced linear algebra” in the title, have appeared in the last few years. These include Yang’s *A Concise Text on Advanced Linear Algebra*, Loehr’s *Advanced Linear Algebra*, Cooperstein’s *Advanced Linear Algebra*, Roman’s *Advanced Linear Algebra*, Weintraub’s *A Guide to Advanced Linear Algebra*, and Lax’s *Linear Algebra and its Applications*.

The book now under review is Helene Shapiro’s vision of what a second course in the subject should look like. (Or perhaps I should say second and third course, since there is more than a semester’s worth of material in it.) It offers substantial differences from the competition and has much to recommend it.

Consistent with the book’s origins (we are told that it evolved from lecture notes prepared by the author for two different courses, one on advanced linear algebra and the other on combinatorial matrix theory), it has a distinctly combinatorial slant to it, but covers enough topics to be quite versatile and accommodate a wide range of preferences. It is also quite well written, strikes an excellent balance between theory and application, and contains both a good supply of examples and a nice assortment of exercises. Based on a quick perusal, it did not appear to me that many of the exercises were terribly difficult, so most should be well within the capabilities of a senior undergraduate. Solutions to the exercises are not provided, which I view as a pedagogical plus.

The first three chapters comprise about fifty pages and largely (particularly in chapter 1) address topics that are discussed in introductory courses, albeit perhaps not at the level of generality involved here. Chapter 1 covers the basic material on vector spaces, matrices and linear transformations; the author works, however, with vector spaces over arbitrary fields, not just the fields of real and complex numbers. In one other respect, however, the book does *not* generalize the contents of a first course: the term “basis” is defined only for finite-dimensional spaces, and the existence of a basis for such a space is simply recalled as an assumed fact, not reproved here.

Chapter 2 introduces inner product spaces and chapter 3 discusses eigenvalue theory. Even in these early chapters, however, there are sections on a number of topics that may not get covered in a first course, including: partitioned and block matrices, tensor products, diagonalization and triangularization, the QR factorization, Gershgorin circles, linear functionals and the dual space.

Canonical forms are the subjects of the next chapter, which is unusual in several respects. First, the proof of the existence of the Jordan Canonical form is not one generally found in the textbook literature; the author credits it to a talk by Paul Halmos at a meeting of the International Linear Algebra Society that she attended. The proof is fairly long, as is to be expected, but the author helps put the students at ease by pointing out that anybody “feeling a bit dazed” by it “should take heart: in most cases where the Jordan canonical form is used, it suffices to know what it is — i.e., what the block diagonal structure looks like.” Second, in addition to the Jordan Canonical form, the text covers the less well-known Weyr Canonical form. This is rarely discussed in the textbook literature; the only other book that I could conveniently find it in is the second (but not the first) edition of *Matrix Analysis* by Horn and Johnson. The rational canonical form, it should be noted, is not discussed in this text.

The next six chapters all involve matters related to inner products and norms. One topic of considerable importance here is operators and matrices acting on such spaces, and the author does a good job with this, devoting a chapter each to normal and Hermitian matrices and discussing both in considerable detail. (Unitary matrices were defined way back in chapter 2, but unitary similarity is discussed here.) Particularly nice features of these chapters include a theorem giving eight different equivalent characterizations of a normal matrix (and a reference to an article giving many more), proofs of both the Rayleigh-Ritz and Courant-Fischer theorems, and the singular value decomposition discussed from several viewpoints (first limited to nonsingular matrices, using the polar decomposition, and then for arbitrary matrices).

Other topics covered in these chapters include matrix norms, Householder transformations, eigenvalue computation, numerical range, and simultaneous triangulation. Some of the topics here would not be out of place in a numerical linear algebra course, but such a course would typically include many topics (e.g., efficiency of algorithms) not treated here.

Chapters 11 through 16 are quite combinatorial. There are chapters here devoted to such traditional combinatorial topics as graphs, digraphs, and block designs, and also linear algebra topics that have a combinatorial slant (e.g., matrices with only 0 and 1 as entries, circulant matrices, Hadamard matrices).

In some of these chapters, the combinatorics seems to dominate the linear algebra. The chapter on block designs and finite projective planes, for example, is so heavily combinatorial in nature, that, reading it, I sometimes wondered when (or if) any sophisticated linear algebra would appear. It did, at the end: the theory of quadratic forms is used to prove the Bruck-Ryser-Chowla theorem, itself a fairly sophisticated combinatorial result that even some combinatorics texts omit the proof of altogether (e.g., Beeler’s *How to Count*) or prove only in part (e.g., Mazur’s *Combinatorics: A Guided Tour*).

Chapter 17 is on nonnegative matrices, including, of course, Perron-Frobenius theory. Although this is a topic that is not generally treated in combinatorics textbooks, parts of this chapter make use of some of the material found in chapters 11 (block cycle matrices) and 16 (directed graphs).

The remaining two chapters explore two significant applications of linear algebra in some detail. Chapter 18 is on error-correcting codes and provides a nice introduction to the subject. The linear algebra used here is not very sophisticated but does serve to illustrate why the author’s decision (referred to earlier) to work with vector spaces over arbitrary fields was a useful one; the relevant field here is the field \(\mathbb{F}_2\) of two elements.

Chapter 19 discusses linear dynamical systems; both continuous and discrete models are considered, sometimes in parallel, which I thought was a nice touch. The chapter ends with a section on Markov chains, which uses some results from chapter 17 (nonnegative matrices) and which could probably be covered directly after that chapter if an instructor is pressed for time.

Linear algebra is a great branch of mathematics, and it deserves to be the subject of excellent books. This text — which combines enthusiasm, clear writing, and a distinctive point of view — qualifies.

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Mark Hunacek ([email protected]) teaches mathematics at Iowa State University.