Matrix Methods: Applied Linear Algebra

This is the third edition of a gentle introduction to matrix algebra and matrix methods that includes a modest amount of material on applications. Although “linear algebra” appears in the subtitle, this is not a linear algebra text in the manner of Lay’s or Strang’s books. For example, nowhere in the current book could I find the phrase “linear transformation” or its equivalent. Nor is the word “basis” mentioned, nor any of the usual results about bases. Instead, this is a book that limits itself to matrix algebra, some matrix calculus, and applications thereof. It has a bit of an old-fashioned feel to it.

The authors are clear about their intentions: “This edition remains a textbook for the student, not the instructor. It remains a book on methodology rather than theory.” Proofs are provided as part of the main body of the text “only if they are easy to follow and revealing.”

The book begins with chapters on the basics of matrix and vector operations and systems of linear equations. Concepts of rank and linear independence are introduced in the context of linear systems. There is a full chapter on matrix inverses that includes a short section on LU decomposition. The first major application is called an introduction to optimization, but it is really limited to linear programming and the simplex method.

There is a full chapter on determinants, followed by another on eigenvalues and eigenvectors. The latter includes a section for computing eigenvalues by the power method. The two chapters that follow require calculus. The first of these introduces some matrix calculus, and spends several pages discussing and computing the exponential of a matrix. The next chapter then takes up linear ordinary differential equations of order n, their reduction to systems of first order equations, and solution methods. This goes as far as a discussion of the fundamental matrix for linear homogeneous systems.

Probability and Markov chains are discussed in a disappointingly brief chapter. (Surely, in book touted for its applications, the authors could have included more examples.) Finally, the last chapter focuses on inner products and the least squares method in data analysis, and leads the reader through projections and QR decompositions.

The strong points of the text are the many carefully worked-out numerical examples. This, together with the gently-paced exposition, would make the book a good candidate for self-study. A high school student with some calculus could manage quite well with this book. At the college-level, the book appears to be aimed at science, engineering and economics students, under the (possibly dubious) assumption that, for them, knowing methods is more important than learning theory. There is a very brief appendix on technology, notable only for the opportunity it gives the authors to emphasize their belief that a first course should focus on concepts, not “number crunching”.

Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.