Basics of Matrix Algebra for Statistics with R (Chapman & Hall/CRC

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**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.

"...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

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