Reviews of Books and Teaching Materials

Nick Fieller’s Basics of Matrix Algebra for Statistics with R is a well-written book and organized well for readers of an introductory book on linear algebra. In its first seven chapters, it covers the basics of vectors and matrices, rank, determinants, inverse, eigenanalysis, calculus of vectors and matrices. The eighth chapter covers further topics in linear algebra (QR, LU, and LDU decompositions, generalized inverses, Hadamard and Kronecker products) and the final chapter discusses important applications of linear algebraic results in statistics. Each chapter concludes with exercises, and solutions are provided. The author provides a thorough introduction to the statistical software R at the beginning of the first chapter, as well as many R examples with output throughout the book. Unfortunately, the book falls short in certain areas. In its title, its preface (p. xvii), and several places throughout the text, the author emphasizes the linear algebraic results needed for statistics. However, despite efforts in that direction (e.g., dealing mostly with real and symmetric matrices), the book is not fine-tuned for the statistical reader, both from a substantive and a technical details point of view. In the introduction to chapter 1, the author claims that “the aim of this book is to provide a guide to elementary matrix algebra sufficient for undertaking intermediate and advanced statistical courses such as multivariate data analysis and linear models. ” While this is true for some parts of the book where the reader is introduced to relevant statistical applications, opportunities to explain other fundamental statistical ideas are missed, particularly: The bookdoes not discuss vector spaces at all, but this topic is pivotal to understanding the geometry of linear models. Hilbert spaces are also important in understanding probability, projections, and conditional expectation. When discussing the determinant of partitioned matrices (chapter 4), no statistical motivation (e.g., conditional distributions derived from a multivariate normal distribution) is provided. The same thing applies to the inversion of partitioned matrices in chapter 5 (section 5.5), where a number of formulas for inverting partitioned matrices are provided but there is no discussion of their usefulness in statistical derivations and/or contexts. The book lacks a conceptual discussion of eigenvalues and eigenvectors. This topic is extremely important in statistical analysis and in understanding the structure of a covariance matrix. If matrix operations were presented as transformations, eigenvectors and positive/negative definite matrices could have been motivated in a more conceptual way, which would be helpful for readers with less mathematical background. Anothermajor statistical application of eigenanalysis is multivariate confidence regions based on a maximum likelihood estimate. This also could have been used to motivate eigenanalysis for readers with statistical interests. The quadratic forms and the Rayleigh quotient (p. 91) are often used in multivariate statistics, particularly in principal component analysis. Given the scope of the book,motivating these seemingly abstract topics through statistical ideas would have been helpful. The matrix exponential (p. 94) is an isolated topic in the book. Mentioning phase-type distributions and the special case of the exponential distribution would have provided a statistical application. Discussing the square root of a matrix (p. 94) provides an excellent opportunity to connect to the ideas of statistical distance, clustering, and classification. The reader, however, has to wait until chapter 9 (pp. 143–144) to see a statistical application of thematrix square root, this one in the context of the multivariate normal distribution. Although the author highlights linear algebraic concepts through a number of interesting statistical topics in the last chapter, a lack of concrete examples makes it hard for the reader to fully absorb this material. The book also misses the opportunity to highlight more modern statistical topics. Connections to regularization methods such as ridge and LASSO regression could be made while discussing spectral analysis and/or eigenanalysis, or they could be addressed in the last chapter as a separate subsection. Despite its shortcomings the book would make a good reference for instructors wanting a resource for R implementations of linear algebraic algorithms or for a supplemental text with extra problems and solutions.