Matrix Algebra for Linear Models

**A self-contained introduction to matrix analysis theory and applications in the field of statistics** Comprehensive in scope, __Matrix Algebra for Linear Models__ offers a succinct summary of matrix theory and its related applications to statistics, especially linear models. The book provides a unified presentation of the mathematical properties and statistical applications of matrices in order to define and manipulate data. Written for theoretical and applied statisticians, the book utilizes multiple numerical examples to illustrate key ideas, methods, and techniques crucial to understanding matrix algebra’s application in linear models. __Matrix Algebra for Linear Models__ expertly balances concepts and methods allowing for a side-by-side presentation of matrix theory and its linear model applications. Including concise summaries on each topic, the book also features: * Methods of deriving results from the properties of eigenvalues and the singular value decomposition * Solutions to matrix optimization problems for obtaining more efficient biased estimators for parameters in linear regression models * A section on the generalized singular value decomposition * Multiple chapter exercises with selected answers to enhance understanding of the presented material __Matrix Algebra for Linear Models__ is an ideal textbook for advanced undergraduate and graduate-level courses on statistics, matrices, and linear algebra. The book is also an excellent reference for statisticians, engineers, economists, and readers interested in the linear statistical model. Matrix methods have evolved from a tool for expressing statistical problems to an indispensable part of the development, understanding, and use of various types of complex statistical analyses. This evolution has made matrix methods a vital part of statistical education. Traditionally, matrix methods are taught in courses on everything from regression analysis to stochastic processes, thus creating a fractured view of the topic. Linear Algebra for Linear Models offers readers a unique, unified view of matrix analysis theory (where and when necessary), methods, and their applications. Written for future statisticians, both theoretical and applied, this book emphasizes the key topics that are needed in a concise and accurate way. Emphasis is on understanding and interpreting principal components as an eigenvalue, generalized inverses, and singular value decomposition. The derivation of important results in Analysis of Variance (ANOVA) is made elegant by the use of some of the properties of quadratic forms, the Kronecker product, and special matrices. A large number of numerical examples and exercises are included to further illustrate the motivation behind the concepts Matrix methods have evolved from a tool for expressing statistical problems to an indispensable part of the development, understanding, and use of various types of complex statistical analyses. This evolution has made matrix methods a vital part of statistical education. Traditionally, matrix methods are taught in courses on everything from regression analysis to stochastic processes, thus creating a fractured view of the topic. Matrix Algebra for Linear Models offers readers a unique, unified view of matrix analysis theory (where and when necessary), methods, and their applications. Written f 4.4 Direct Product of Matrices4.5 An Important Property of Determinants; 4.6 The Trace of a Matrix; 4.7 Matrix Differentiation; 4.8 The Least Square Estimator Again; 4.9 Summary; Exercises; Section 5 Vector Spaces; 5.1 Introduction; 5.2 What Is a Vector Space?; 5.3 The Dimension of a Vector Space; 5.4 Inner Product Spaces; 5.5 Linear Transformations; 5.6 Summary; Exercises; Section 6 The Rank of a Matrix and Solutions to Systems of Equations; 6.1 Introduction; 6.2 The Rank of a Matrix; 6.3 Solving Systems of Equations with Coefficient Matrix of Less than Full Rank; 6.4 Summary; Exercises Part II Eigenvalues, the Singular Value Decomposition, and Principal ComponentsSection 7 Finding the Eigenvalues of a Matrix; 7.1 Introduction; 7.2 Eigenvalues and Eigenvectors of a Matrix; 7.3 Nonnegative Definite Matrices; 7.4 Summary; Exercises; Section 8 The Eigenvalues and Eigenvectors of Special Matrices; 8.1 Introduction; 8.2 Orthogonal, Nonsingular, and Idempotent Matrices; 8.3 The Cayley-Hamilton Theorem; 8.4 The Relationship between the Trace, the Determinant, and the Eigenvalues of a Matrix; 8.5 The Eigenvalues and Eigenvectors of the Kronecker Product of Two Matrices 2.4 Solution to Linear Equations Using Determinants2.5 Gauss Elimination; 2.6 Summary; Exercises; Section 3 The Inverse of a Matrix; 3.1 Introduction; 3.2 The Adjoint Method of Finding the Inverse of a Matrix; 3.3 Using Elementary Row Operations; 3.4 Using the Matrix Inverse to Solve a System of Equations; 3.5 Partitioned Matrices and Their Inverses; 3.6 Finding the Least Square Estimator; 3.7 Summary; Exercises; Section 4 Special Matrices and Facts about Matrices That Will Be Used in the Sequel; 4.1 Introduction; 4.2 Matrices of the Form aIn+bJ n; 4.3 Orthogonal Matrices Matrix Algebra for Linear Models; Copyright; Contents; Preface; Acknowledgments; Part I Basic Ideas about Matrices and Systems of Linear Equations; Section 1 What Matrices Are and Some Basic Operations with Them; 1.1 Introduction; 1.2 What Are Matrices and Why Are They Interesting to a Statistician?; 1.3 Matrix Notation, Addition, and Multiplication; 1.4 Summary; Exercises; Section 2 Determinants and Solving a System of Equations; 2.1 Introduction; 2.2 Definition of and Formulae for Expanding Determinants; 2.3 Some Computational Tricks for the Evaluation of Determinants 8.6 The Eigenvalues and the Eigenvectors of a Matrix of the Form aI + bJ8.7 The Loewner Ordering; 8.8 Summary; Exercises; Section 9 The Singular Value Decomposition (SVD); 9.1 Introduction; 9.2 The Existence of the SVD; 9.3 Uses and Examples of the SVD; 9.4 Summary; Exercises; Section 10 Applications of the Singular Value Decomposition; 10.1 Introduction; 10.2 Reparameterization of a Non-full-Rank Model to a Full-Rank Model; 10.3 Principal Components; 10.4 The Multicollinearity Problem; 10.5 Summary; Exercises Section 11 Relative Eigenvalues and Generalizations of the Singular Value Decomposition