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# Basics of Matrix Algebra for Statistics with R

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## Book Description

*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.

## Table of Contents

**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.*

## Author(s)

### Biography

**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.

## Reviews

"…belongs to the category of mathematics books that integrate a programming language with substantive content. On the substantive side, the author has meticulously selected matrix algebra topics that are fundamental to learning, using, and understanding statistics. In this manner, the reader is saved time by focusing on matrix mathematics which is of most relevance to statistics. In addition, an instructor also benefits from the concise introduction to matrix algebra related to statistics. Therefore, this book can easily be adopted as a matrix algebra supplemental book in a syllabus on statistics. The exercises are short but rigorous, with detailed solutions provided at the end of the book...as a traditional text to teach practical matrix algebra to students taking multivariate and more advanced statistics courses, this book can be of good use."

—Abdolvahab Khademi, University of Massachusetts,Journal of Statistical Software, July 2016"Key features of the book include highlighting useful tricks when manipulating matrices, derivation of key results with step-by-step cross-referenced explanations and demonstrations of implementing the techniques in R using numerical examples…it is a good beginner’s guide to understanding and manipulating matrices in R. It is suitable for early year undergraduate students and anyone who wishes to be introduced to matrix algebra in R in preparation for high-level or specialised studies in statistics. The book’s collection of summaries and key results also make it a good handbook for any statistician to refer to."

—Shuangzhe Liu,Stastistical Papers, July 2016"… 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. … 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 … it is very easy to find the R command for carrying out a specific matrix calculation. … 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. … the inclusion of exercises facilitates the use of this book as a course text."

—MAA Reviews, January 2016