In the text, *Matrix Theory and Applications for Scientists and Engineers*, Alexander Graham’s goal is to bridge what he claims is a gap between matrix theory texts that are rich in rigor but are not at the appropriate level for the traditional scientist who would like to understand techniques of linear algebra and the texts that offer the various techniques with a minimal view of underlying principles or justifications. This book is designed to be a middle of the road book that helps the student to understand the concepts with keeping a certain bar of rigor, notation, and applications for scientists, economists, and engineers. Specific focus is given to control theory in engineering in Chapter 8, the final chapter of the book.

Graham organizes the book into 8 chapters starting with Matrices, Vector Spaces, Linear Transformations, The Rank and the Determinant of a Matrix, Eigenvalues and Eigenvectors, Canonical Forms and Matrix Functions, and Inverting a Matrix. Chapters 6 and 7 on Eigenvalues and Eigenvectors and Canonical Forms and Matrix Functions are the most important for engineering students as they discuss obtaining the decomposition of a vector space and choosing a basis relative to which a linear transformation has a block-diagonal matrix representation.

The text covers the basics of matrix theory but weaves in more advanced topics quickly. Chapter 1 introduces the student to a matrix being a rectangular array of numbers but then goes on to point out how the elements belong to a field \( F \) so that \( A \) is a matrix over the field \( F \). The chapter also includes an introduction to set theory notation to be used in the whole book. Other advanced topics found in Chapter 1 include the definitions of orthogonal, unitary, conjugate, Hermitian, and skew-Hermitian.

Professors and textbook writers will be surprised to see the layout of certain topics in this text. Personally, I like how Graham puts matrix algebra as the first chapter but, he doesn’t discuss systems of linear equations, homogenous systems, elementary row operations, and row reduction processes until later in the text (Chapter 5 and 8 respectively.) This is quite the opposite of the norm. Some readers may ask why and others may like this approach. However, I’m puzzled. For example, on pages 48-49, Example 2.3 asks the reader to write \( z=(-2,1,3) \) as a linear combination of the following vectors, \( y_{1}=(2,1,0) \), \( y_{2}=(-1,2,1) \), and \( y_{3}=(1,0,1) \). The system of linear equations is: \( -2 = 2 \alpha – \beta + \gamma \), \( 1 = \alpha + 2\beta \), and \( 3 = \beta + \gamma \). Graham writes the solution as \( \alpha = -4/3 \), \( \beta = 7/6 \), and \( \gamma = 11/6 \). I believe this would be a great place to use an augmented matrix and carry on the ideas of matrix algebra found in Chapter 1.

Section 7.5 discusses the decomposition of a vector space into A-invariant subspaces by constructing a matrix P so that \( P^{-1}AP \) is in block diagonal form. This is discussed using a minimal polynomial for A and by Theorem 7.12 on page 197, letting \( m \) be the minimal polynomial for A, assume that the product of two relatively prime monic polynomials \( f \) and \( g \) so that \( m(x) = f(x)g(x) \), then \( V=W_{1} \oplus W_{2} \) where \( W_{1}=\mbox{ker}[g(A)] \) and \(W_{2}=\mbox{ker}[f(A)] \). The examples in this section are great; especially Example 7.10, however, a reference to how this can be applied would only enhance the section.

The final chapter makes this book unique among matrix theory texts as it is focused on the practical (not numerical but analytical) methods and reasoning for inverting matrices. The purpose of this section is central to Control Engineering. The chapter starts by re-considering the system of equations:

\[ \dot{x}=Ax+Bu \]

\[ \dot{y}=Cx \]

By taking Laplace transformations, the resulting matrix is \( C(sI – A)^{-1}B \), which is the transfer function matrix relating the input and output of the system. This is also the same chapter where Graham introduces the three elementary row operations, which gives this book an interesting sequence of topics. However, the chapter does move forward with more advanced topics such as The Inverse of a Vandermonde Matrix, Faddeeva’s Method, and Inverting a Matrix with Complex Numbers.

In reading the Preface, I was excited to see how the final chapter would be written for the application of Control Engineering. However, the text did not present how the transfer function matrix is applied, which was disappointing. Teaching engineering students for the past nine years, I can personally say that the students want to see an example, or two, of how the mathematics can be applied. While this text has some great examples with a full set of answers to all problems in the back of the book, there are some parts that fall short and other parts are in a non-traditional order, which may disappoint some professors considering required texts for their classes. I can also see how this text may leave a student asking for more information.

Peter Olszewski is a Mathematics Lecturer at The Pennsylvania State University, The Behrend College, an editor for Larson Texts, Inc. in Erie, PA, and is the 362nd Chapter Advisor of the Pennsylvania Alpha Beta Chapter of Pi Mu Epsilon. His Research fields are in mathematics education, Cayley Color Graphs, Markov Chains, and mathematical textbooks. 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 and bass, reading, gardening, traveling, and painting landscapes.