A Polynomial Approach to Linear Algebra

Anyone who has studied linear algebra knows that the minimal and characteristic polynomials of a matrix play an important role in the development of the theory. This interesting and unusual text shows that the connections between linear algebra and polynomials go far deeper than this. The central idea of this book is to use a certain kind of operator, called a shift operator, whose definition (sketched below) depends on a given polynomial, as a tool for studying linear algebra and its applications.

After an introductory chapter quickly reviewing algebraic preliminaries (more than a typical undergraduate semester’s worth of groups, modules, rings and fields, including things like Euclidean domains and examples like the ring of truncated Laurent series) there are three chapters on vector spaces, determinants and linear transformations. The discussion here is (for the most part) fairly conventional in its choice of topics, and is succinct, elegant and efficient; it takes only about six pages to go from the definition of a vector space through proofs of the usual facts about finite bases (every linearly independent set can be extended to a basis, any two finite bases have the same number of elements, etc.). Although the shift operators have not yet been defined in these chapters, polynomials are never far from the surface here, and topics such as Lagrange interpolation and the Sylvester Resultant appear.

Things start to become less conventional in chapter 5, which introduces the aforementioned shift operators, which are defined as follows: for q(z) a monic polynomial of degree n with coefficients from a field F, the vector space X_q is defined to be to be the space of all possible remainders when a polynomial is divided by q; X_q is, therefore, just the vector space of all polynomials of degree less than n. The shift operator S_q is the mapping from X_q to itself defined by taking S_q(f) to be the remainder when zf(z) is divided by q. Properties of the mapping S_q (and other operators called the positive and negative shifts) are studied, then in chapter 6 this machinery is applied to study the structure of linear transformations, including the Jordan canonical form.

Perhaps it is simply a matter of personal taste, but I found the notation to be rather dense and sometimes unusual, and on more than one occasion I found myself flipping back in the book to remind myself what something meant. This notational density may be inevitable given the subject matter, but I would certainly recommend, as an improvement for any future editions, the inclusion of a notations index.

The question also arises, of course, whether this “shift operator approach” is the best way to learn topics like the Jordan canonical form. I found the discussion interesting, but then again I already knew what the Jordan form was; I’m not sure this is the best way to approach the subject for people learning it for the first time. The Jordan form is sufficiently difficult to grasp that I wonder whether adding these new concepts is really a pedagogical improvement over the more traditional approaches to the subject.

Chapter 7, on inner product spaces, marks a return to more familiar material. The exposition here does not rely on the shift operators defined previously but does manage, in the space of about 30 pages, to give not only a fairly complete overview of the basic theory of inner product spaces (real and complex spaces treated simultaneously), but also a number of additional topics: the mini-max theorem for calculating eigenvalues (in terms of the Rayleigh quotient) for self-adjoint operators, the Cayley transform, partial isometries, polar form, singular values and singular vectors, positive operators, and unitary embeddings.

The author states in the preface that he has used these seven chapters for a year-long course in linear algebra at Ben Gurion University in Israel. He doesn’t mention whether this was an undergraduate or graduate course, but it seems clear to me that these chapters are pitched at much too high a level for most undergraduate linear algebra courses in the United States. The discussion here is obviously vastly more sophisticated, for example, than what is found in the standard introductory texts like Lay’s Linear Algebra and its Applications, and is even more demanding of a reader than is Hoffman and Kunze’s classic Linear Algebra, if for no other reason than that it requires a fairly good background in abstract algebra to understand.

Chapter 8 of the text introduces forms (quadratic, sesquilinear and bilinear). After a general discussion of them, the polynomial-theoretic ideas that were developed earlier are applied to give a detailed discussion of examples, specifically the Hankel and Bezoutian forms. The chapter then concludes with tensor products, both of vector spaces and modules, including as an important special case tensor products where the underlying ring consists of polynomials.

The remaining four chapters of the text seem somewhat more specialized and geared to applications. There is a chapter on stability theory, linear systems theory, Hardy spaces and Hankel norm approximation theory. Here again, I think the author is being overly optimistic when he states that the material in the last chapter is “very much within the grasp of a well motivated undergraduate,” but certainly these topics are not part of a standard linear algebra text (even at the graduate level) and help make this book a particularly useful source of information for people interested in them.

A review of this book on amazon.com comments on what the reviewer thought to be an “absolutely astounding” number of typos and errors. This is, however, apparently a review of an earlier edition of the book (although it appears as a review of the second edition), since I didn’t notice anywhere near that many (and the particular error mentioned by the reviewer does not appear in my edition). However, I did catch some, ranging from trivial grammatical typos (“a zerodivisors”) to incorrect use of symbols (the multiplicative identity of a ring, denoted 1 in one sentence, suddenly becomes e in the next) to statements of theorems that omit necessary (albeit nit-picky) hypotheses (such as the fact that a certain subset must be nonempty). In addition, there are occasional uses of terminology that struck me as decidedly nonstandard: the coset aH in a group G is called a right coset and Ha is called a left one, for example. All this was relatively minor, I thought, and did not detract greatly from my enjoyment of this interesting text.

This is certainly a linear algebra text with a distinct personality all its own. I cannot recommend it for an undergraduate course, but it can be used with profit by people interested in the special topics at the end (say, in a topics course, or self-study), or by good students in a graduate linear algebra course with a solid background in abstract algebra who are interested in learning the material in a rather nonstandard way.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.