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Basics of Matrix Algebra for Statistics with R

Nick Fieller
Chapman & Hall/CRC
Publication Date: 
Number of Pages: 
The R Series
[Reviewed by , on

Nick Fieller’s book Basics of Matrix Algebra for Statistics with R provides a concise and straightforward presentation of matrix algebra techniques that are commonly used in statistics. Furthermore, the book discusses how to implement numerical instances of these techniques using R. I would describe this book as a compendium of definitions and results on matrix algebra and R commands for carrying out corresponding calculations. If you have a need or desire to carry out matrix computations in R, then it is likely that here you will find the needed commands.

There are several nice features of Basics of Matrix Algebra for Statistics with R. Because of the organization of the book, it is very easy to find the R command for carrying out a specific matrix calculation. First off, the index is good. Second, there is a summary of many useful commands in the first chapter. Finally, in each section of the text, the R commands clearly stand out. These features make the book useful as a reference. In addition, the author provides helpful tips and tricks for working with R.

Another positive feature of this book is the applications to statistics. While many of the applications are mixed in with the chapters on specific matrix algebra techniques, there is also a single chapter devoted entirely to statistical applications.

Even though Basics of Matrix Algebra for Statistics with R appears most useful as a reference or quick guide for statistics related matrix calculations, the inclusion of exercises facilitates the use of this book as a course text. Each chapter contains a good number of exercises' solutions are outlined at the end of the text. There are both computational exercises and exercises of a more theoretical nature.

Finally, it is natural to compare this book with texts on applied or numerical linear algebra such as Matrix Computations by Golub and Van Loan or Matrix Analysis by Horn and Johnson. While Basics of Matrix Algebra for Statistics with R does share some content with such books, its aim is completely different. For instance, it does not develop a great deal of general theory, focusing rather on specific tricks and techniques that are common in statistical calculations involving matrices. Consider as an example Chapter 6, which discusses eigenvalues and eigenvectors but in a restricted way that is nevertheless appropriate in statistical applications. There are many other examples of this in the text. All things considered, Basics of Matrix Algebra for Statistics with R is conveniently organized, well-written and should prove very useful for the purposes it was designed for.

Jason M. Graham is an assistant professor in the department of mathematics at the University of Scranton, Scranton, Pennsylvania. His current professional interests are in teaching applied mathematics and mathematical biology, and collaborating with biologists specializing in the collective behavior of groups of organisms.

Further Reading
Guide to Notation
An Outline Guide to R
Inputting Data to R
Summary of Matrix Operators in R
Examples of R Commands


Vectors and Matrices
Matrix Arithmetic
Transpose and Trace of Sums and Products
Special Matrices
Partitioned Matrices
Algebraic Manipulation of matrices
Useful Tricks
Linear and Quadratic Forms
Creating Matrices in R
Matrix Arithmetic in R
Initial Statistical Applications



Rank of Matrices
Introduction and Definitions
Rank Factorization
Rank Inequalities
Rank in Statistics


Introduction and Definitions
Implementation in R
Properties of Determinants
Orthogonal Matrices
Determinants of Partitioned Matrices
A Key Property of Determinants




Introduction and Definitions
Implementation in R
Inverses of Patterned Matrices
Inverses of Partitioned Matrices
General Formulae
Initial Applications Continued


Eigenanalysis of Real Symmetric Matrices
Introduction and Definitions
Implementation in R
Properties of Eigenanalyses
A Key Statistical Application: PCA
Matrix Exponential
Eigenanalysis of Matrices with Special Structures
Summary of Key Results




Vector and Matrix Calculus
Differentiation of a Scalar with Respect to a Vector
Differentiation of a Scalar with Respect to a Matrix
Differentiation of a Vector with Respect to a Vector
Differentiation of a Matrix with Respect to a Scalar
Use of Eigenanalysis in Constrained Optimization




Further Topics
Further Matrix Decompositions
Generalized Inverses
Hadamard Products
Kronecker Products and the Vec Operator




Key Applications to Statistics
The Multivariate Normal Distribution
Principal Component Analysis
Linear Discriminant Analysis
Canonical Correlation Analysis
Classical Scaling
Linear Models



Outline Solutions to Exercises