Basics of Matrix Algebra for Statistics with R

A Thorough Guide to Elementary Matrix Algebra and Implementation in R Basics of Matrix Algebra for Statistics with R provides a guide to elementary matrix algebra sufficient for undertaking specialized courses, such as multivariate data analysis and linear models. It also covers advanced topics, such as generalized inverses of singular and rectangular matrices and manipulation of partitioned matrices, for those who want to delve deeper into the subject. The book introduces the definition of a matrix and the basic rules of addition, subtraction, multiplication, and inversion. Later topics include determinants, calculation of eigenvectors and eigenvalues, and differentiation of linear and quadratic forms with respect to vectors. The text explores how these concepts arise in statistical techniques, including principal component analysis, canonical correlation analysis, and linear modeling. In addition to the algebraic manipulation of matrices, the book presents numerical examples that illustrate how to perform calculations by hand and using R. Many theoretical and numerical exercises of varying levels of difficulty aid readers in assessing their knowledge of the material. Outline solutions at the back of the book enable readers to verify the techniques required and obtain numerical answers. Avoiding vector spaces and other advanced mathematics, this book shows how to manipulate matrices and perform numerical calculations in R. It prepares readers for higher-level and specialized studies in statistics.

Introduction Objectives 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 Vectors 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 Determinants Introduction and Definitions Implementation in R Properties of Determinants Orthogonal Matrices Determinants of Partitioned Matrices A Key Property of Determinants Inverses Introduction and Definitions Properties Implementation in R Inverses of Patterned Matrices Inverses of Partitioned Matrices General Formulae Initial Applications Continued Eigenanalysis of Real Symmetric Matrices Introduction and Definitions Eigenvectors Implementation in R Properties of Eigenanalyses A Key Statistical Application: PCA Matrix Exponential Decompositions Eigenanalysis of Matrices with Special Structures Summary of Key Results Vector and Matrix Calculus Introduction 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 Introduction Further Matrix Decompositions Generalized Inverses Hadamard Products Kronecker Products and the Vec Operator Key Applications to Statistics Introduction The Multivariate Normal Distribution Principal Component Analysis Linear Discriminant Analysis Canonical Correlation Analysis Classical Scaling Linear Models Outline Solutions to Exercises Bibliography Index Exercises appear at the end of each chapter.

Dr. Nick Fieller is a retired senior lecturer in the School of Mathematics and Statistics and an honorary research fellow in archaeology at the University of Sheffield. His research interests include multivariate data analysis and statistical modeling in the pharmaceutical industry, archaeology, and forensic sciences.