In *Applied Matrix Algebra*, Lawrence Harvill has placed great emphasis on applications rather than theoretical concepts. He has made an effort to motivate students, showing them how matrix algebra is used in engineering, computer science, physics, and across various disciplines. Students are not bogged down with traditional questions such as asking them to show that something is a subspace or that a function is a linear transformation. This text is all about applications, which makes students realize what they will encounter later on in either undergraduate or graduate programs.

Before heading into applications, the author develops an understanding of matrix algebra and recalls basic principles of first and second semester Calculus, least squares, steady state linear circuits, input-output economic analysis, the solution of potential problems, difference equations, random walks, gambling problems, and the analysis of plane trusses, from which many applications problems are derived.

The first two chapters are very similar to most other applied matrix algebra texts. With the introduction of the term *matrix*, first coined by James Sylvester in 1850, Harvill captures the interest of the student and fellow professors. The fundamentals of matrix algebra, linear algebraic equations and vectors, applications, and finally, the determinant are presented in sequence. In any matrix algebra text, eigenvalues and eigenvectors should follow directly after the discussion of the determinant, which Harvill does nicely.

The rest of the text goes further into applications to numerical analysis: LU factorizations, Cholesky’s method, and Householder transformations, just to name a few. As technology has advanced over the years, mathematicians, engineers, and scientists, have come to grips with the usefulness of computation. These advances have certainly made us look at matrices in a new, and better, light. For the applications, various technological tools are used to give the student a visual understanding of how programs present matrices and solutions. Microsoft *Excel*, *Matlab*, and *Mathematica* are used for presentations throughout the book.

Students whose classes include topics from chapters III and VI–IX will gain a better and more sophisticated understanding of complex problems. One of my favorites is chapter VIII, “Applications of the Difference and Differential Equations.” In example 1, Harvill describes the classic pipe-flow problem, which all differential equation students come across in a traditional ODE course. For those students who are on the electrical engineering path, example 4 describes the solution of a classic electrical circuit problem. Connections are made with eigenvalues and eigenvectors.

I have a few concerns. Some of the problems and solutions are trimmed off. This is evident on pages 56, 98, 113, 119, and 228, for example. Some of the images of the matrices are condensed, which they shouldn’t be. This is evident on page 136 for exercises 4–7. I would personally, like to see a few more examples of row-reducing matrices using elementary matrices. This is a very important and fundamental skill that the author should consider more in detail through examples. In order to conform to the usual notation, eigenvalues should be denoted λ. Perhaps color images would be more useful for the students, for example in the images on pages 149, 150, and 285.

*Applied Matrix Algebra*’s intended audiences are mathematicians, engineers, and scientists. The book does a nice job of getting the topics across to such readers, who should have some background in matrices and a basic understanding of abstract algebra in order to fully appreciate the text. But one can easily relearn from this book whatever the reader doesn’t recall from previous classes.

Ten years ago, there were no sophisticated algorithms to find solutions to complex problems such as those presented in this book. Now we are able to find solutions with smaller margins of error than ever before. Students should find this book informative and learn to apply what they read to their fields. I see this text on the shelves of student’s libraries to use as a reference even after graduation.

Peter Olszewski is a Mathematics Lecturer at Penn State Erie, The Behrend College and is also an editor for Larson Texts, Inc. in Erie, PA. He can be reached at [email protected]. Webpage: www.personal.psu.edu/pto2. Outside of teaching and textbook editing, he enjoys playing golf, playing guitar, reading, gardening, traveling, and painting landscapes.