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Book Review: Matrix Methods: Applied Linear Algebra and Sabermetrics
Article in AIAA Journal · December 2020
DOI: 10.2514/1.J060417
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Rana Zakerzadeh
Duquesne University
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AIAA JOURNAL
Vol. , No. ,
Book Reviews
BOOK REVIEWS published in this section reflect the opinions of their individual authors. They are not necessarily the opinions of the Editors of
this journal or of AIAA.
Book Review: Matrix Methods: Applied Linear Algebra and Sabermetrics
Richard Bronson and Gabriel B. Costa, Academic Press, London, 2020, 512 pp., $99.95.
DOI: 10.2514/1.J060417; published online XX epubMonth XXXX.
methods are the essence of applied
M ATRIX-DRIVEN
linear algebra with a wide range of applications in
system of simultaneous linear equations. Chapter 6
focuses on eigenvalues and eigenvectors, and their properties. This is done first by introducing analytical methods,
then proceeds by a discussion of the power methods for
computation of the dominant eigenvalues and eigenvectors of large matrices. Chapter 7 is devoted to “Matrix
Calculus” and defines some of the widely used matrix
functions such as polynomials and exponentials, develops
techniques for calculating these functions and their derivatives, and discusses some of their important properties.
This chapter is particularly useful for solving a system of
linear ordinary differential equations where evaluation of
the matrix exponential is necessary. This solution strategy
is discussed further in Chapter 8. In Chapter 9, an overview of the Bernoulli trials and Markov chains using a
matrix-oriented approach is presented. This chapter is
focused on the probability theory and provides insights
into the use of matrices for applications in industry. In
Chapter 10, orthonormal vectors, QR decomposition to
convert a matrix into orthogonal and upper triangular
components, and least squares are discussed. The final
two chapters, 11 and 12, deal with sabermetrics. The term
“sabermetrics” is derived from the acronym SABR, which
stands for the Society for American Baseball Research,
and is for empirical analysis and statistical reasoning about
baseball that measures in-game activity. An introduction to
the development of sabermetrics and an outline of its
fundamentals are provided in Chapter 11, followed by an
illustration on how one construct sabermetrics. For instructors who are interested in offering a course on this subject, a
sample final exam is provided at the end of this chapter.
Chapter 12 highlights several sabermetric measures, followed by several problems and questions. Certain aspects
of specific instruments, and the reason for their value in
sabermetrics are also considered in this chapter. In chapters
11–12, it is assumed that the reader is familiar with the
basics of baseball; otherwise, it would be quite difficult to
follow the materials. I feel that these chapters are not selfcontained and do not include sufficient examples and applications. I would recommend combining these two chapters
into one, since some of the metrics mentioned in Chapter
11 are not defined until later in Chapter 12.
This text provides more than enough materials for
a one-semester course for engineering and science
solving real-world problems. Rapid changes in technology
have made the implementation of matrices relevant to a
broad range of fields. In addition to STEM, linear algebra is
now important in many other disciplines such as medicine,
life sciences, and humanities. The interest on the subject
has been increased significantly in recent years, due to
eminence of modern fields such as machine learning and
quantum computing.
The Matrix Methods was first published nearly 50 years
ago. The current Fourth edition offers a fresh and unique
balance between the theory and the computation of matrices, while highlighting their interdependence. The authors
have done a commendable job of describing basic and
advanced methods to tackle many important problems
involving matrices. This is a comprehensive text covering
matrix theory in a nutshell and is written to help the
pedagogy of linear algebra to keep up with the rapid and
continuously growing importance of this subject. The content is easy to follow for those who have not been previously exposed to this subject, and the level of mathematics
is kept at the undergraduate level. This new edition has the
same general order of topics as in previous versions, followed by two additional chapters that blend the application
of matrix methods with baseball, under the subject of
sabermetrics.
The book is composed of 12 chapters. Chapter 1 provides an insightful overview of the basic concepts in matrix
operations. In Chapter 2, the methods for solving linear
equations are introduced with a section devoted to the
theory of solutions for a simultaneous system of linear
equations. Chapter 3 begins with an introduction to matrix
inversion and ends with a concise description of the application of inverses in solving simultaneous equations.
Chapter 4 covers two classical techniques to optimization,
namely the linear programming and the simplex methods.
The former is a very useful way of solving optimization
problems via objective function. The latter provides an
efficient means of conducting linear programming by hand
using slack variables, tableaus, and pivot variables as a
method to optimization. Chapter 5 deals with methods
to find the determinant of a matrix using cofactors. This
chapter also describes the Cramer’s rule for solving the
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majors. It speaks student’s language by illustrating new
subjects clearly and concisely. There is a richness to the
material that goes beyond most texts at this level. I would
recommend the book especially for Chapters 6–8, as I
found the topics of matrix calculus and the systems of
linear differential equations are covered very nicely with
unique perspectives. Discussion of matrix exponentiation for the solution of differential equations is also very
clearly explained. The book contains a good mixture of
classical and contemporary methods with a range of
problems with varying levels of difficulty. The examples
within the text are thorough, useful, and revealing. Each
chapter includes a reasonably diverse set of exercises. A
carefully prepared solution manual is included at the end
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BOOK REVIEWS
of the book providing answers and/or hints to selected
problems.
Matrix Methods, in my opinion, is a valuable textbook
that someone would want to keep as a reference. It would
be difficult to find another book that addresses such a wide
variety of subjects, ranging from the basics to applications,
in particular statistical reasoning to baseball problems.
Readers who want to explore problems in sabermetrics
and to learn some advanced mathematics and modeling
techniques developed around matrices, will surely enjoy
Matrix Methods.
Rana Zakerzadeh
Duquesne University