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.

Table of Contents:
Preface xiii Acknowledgments xv Part I Basic Ideas about Matrices and Systems of Linear Equations 1 Section 1 What Matrices are and Some Basic Operations with Them 3 1.1 Introduction 3 1.2 What are Matrices and why are they Interesting to a Statistician? 3 1.3 Matrix Notation Addition and Multiplication 6 1.4 Summary 10 Exercises 10 Section 2 Determinants and Solving a System of Equations 14 2.1 Introduction 14 2.2 Definition of and Formulae for Expanding Determinants 14 2.3 Some Computational Tricks for the Evaluation of Determinants 16 2.4 Solution to Linear Equations Using Determinants 18 2.5 Gauss Elimination 22 2.6 Summary 27 Exercises 27 Section 3 The Inverse of a Matrix 30 3.1 Introduction 30 3.2 The Adjoint Method of Finding the Inverse of a Matrix 30 3.3 Using Elementary Row Operations 31 3.4 Using the Matrix Inverse to Solve a System of Equations 33 3.5 Partitioned Matrices and Their Inverses 34 3.6 Finding the Least Square Estimator 38 3.7 Summary 44 Exercises 44 Section 4 Special Matrices and Facts about Matrices that will be used in the Sequel 47 4.1 Introduction 47 4.2 Matrices of the Form aIn + bJn 47 4.3 Orthogonal Matrices 49 4.4 Direct Product of Matrices 52 4.5 An Important Property of Determinants 53 4.6 The Trace of a Matrix 56 4.7 Matrix Differentiation 57 4.8 The Least Square Estimator Again 62 4.9 Summary 62 Exercises 63 Section 5 Vector Spaces 66 5.1 Introduction 66 5.2 What is a Vector Space? 66 5.3 The Dimension of a Vector Space 68 5.4 Inner Product Spaces 70 5.5 Linear Transformations 73 5.6 Summary 76 Exercises 76 Section 6 The Rank of a Matrix and Solutions to Systems of Equations 79 6.1 Introduction 79 6.2 The Rank of a Matrix 79 6.3 Solving Systems of Equations with Coefficient Matrix of Less than Full Rank 84 6.4 Summary 87 Exercises 87 Part II Eigenvalues the Singular Value Decomposition and Principal Components 91 Section 7 Finding the Eigenvalues of a Matrix 93 7.1 Introduction 93 7.2 Eigenvalues and Eigenvectors of a Matrix 93 7.3 Nonnegative Definite Matrices 101 7.4 Summary 104 Exercises 105 Section 8 The Eigenvalues and Eigenvectors of Special Matrices 108 8.1 Introduction 108 8.2 Orthogonal Nonsingular and Idempotent Matrices 109 8.3 The Cayley–Hamilton Theorem 112 8.4 The Relationship between the Trace the Determinant and the Eigenvalues of a Matrix 114 8.5 The Eigenvalues and Eigenvectors of the Kronecker Product of Two Matrices 116 8.6 The Eigenvalues and the Eigenvectors of a Matrix of the Form aI + bJ 117 8.7 The Loewner Ordering 119 8.8 Summary 121 Exercises 122 Section 9 The Singular Value Decomposition (SVD) 124 9.1 Introduction 124 9.2 The Existence of the SVD 125 9.3 Uses and Examples of the SVD 127 9.4 Summary 134 Exercises 134 Section 10 Applications of the Singular Value Decomposition 137 10.1 Introduction 137 10.2 Reparameterization of a Non-full-Rank Model to a Full-Rank Model 137 10.3 Principal Components 141 10.4 The Multicollinearity Problem 143 10.5 Summary 144 Exercises 145 Section 11 Relative Eigenvalues and Generalizations of the Singular Value Decomposition 146 11.1 Introduction 146 11.2 Relative Eigenvalues and Eigenvectors 146 11.3 Generalizations of the Singular Value Decomposition:Overview 151 11.4 The First Generalization 152 11.5 The Second Generalization 157 11.6 Summary 160 Exercises 160 Part III Generalized Inverses 163 Section 12 Basic Ideas about Generalized Inverses 165 12.1 Introduction 165 12.2 What is a Generalized Inverse and how is One Obtained? 165 12.3 The Moore–Penrose Inverse 170 12.4 Summary 173 Exercises 173 Section 13 Characterizations of Generalized Inverses Using the Singular Value Decomposition 175 13.1 Introduction 175 13.2 Characterization of the Moore–Penrose Inverse 175 13.3 Generalized Inverses in Terms of the Moore–Penrose Inverse 177 13.4 Summary 185 Exercises 186 Section 14 Least Square and Minimum Norm Generalized Inverses 188 14.1 Introduction 188 14.2 Minimum Norm Generalized Inverses 189 14.3 Least Square Generalized Inverses 193 14.4 An Extension of Theorem 7.3 to Positive-Semi-definite Matrices 196 14.5 Summary 197 Exercises 197 Section 15 More Representations of Generalized Inverses 200 15.1 Introduction 200 15.2 Another Characterization of the Moore–Penrose Inverse 200 15.3 Still another Representation of the Generalized Inverse 204 15.4 The Generalized Inverse of a Partitioned Matrix 207 15.5 Summary 211 Exercises 211 Section 16 Least Square Estimators for Less than Full-Rank Models 213 16.1 Introduction 213 16.2 Some Preliminaries 213 16.3 Obtaining the LS Estimator 214 16.4 Summary 221 Exercises 221 Part IV Quadratic Forms and the Analysis of Variance 223 Section 17 Quadratic Forms and their Probability Distributions 225 17.1 Introduction 225 17.2 Examples of Quadratic Forms 225 17.3 The Chi-Square Distribution 228 17.4 When does the Quadratic Form of a Random Variable have a Chi-Square Distribution? 230 17.5 When are Two Quadratic Forms with the Chi-Square Distribution Independent? 231 17.6 Summary 234 Exercises 235 Section 18 Analysis of Variance: Regression Models and the One- and Two-Way Classification 237 18.1 Introduction 237 18.2 The Full-Rank General Linear Regression Model 237 18.3 Analysis of Variance: One-Way Classification 241 18.4 Analysis of Variance: Two-Way Classification 244 18.5 Summary 249 Exercises 249 Section 19 More ANOVA 253 19.1 Introduction 253 19.2 The Two-Way Classification with Interaction 254 19.3 The Two-Way Classification with One Factor Nested 258 19.4 Summary 262 Exercises 262 Section 20 The General Linear Hypothesis 264 20.1 Introduction 264 20.2 The Full-Rank Case 264 20.3 The Non-full-Rank Case 267 20.4 Contrasts 270 20.5 Summary 273 Exercises 273 Part V Matrix Optimization Problems 275 Section 21 Unconstrained Optimization Problems 277 21.1 Introduction 277 21.2 Unconstrained Optimization Problems 277 21.3 The Least Square Estimator Again 281 21.4 Summary 283 Exercises 283 Section 22 Constrained Minimization Problems with Linear Constraints 287 22.1 Introduction 287 22.2 An Overview of Lagrange Multipliers 287 22.3 Minimizing a Second-Degree Form with Respect to a Linear Constraint 293 22.4 The Constrained Least Square Estimator 295 22.5 Canonical Correlation 299 22.6 Summary 302 Exercises 302 Section 23 The Gauss–Markov Theorem 304 23.1 Introduction 304 23.2 The Gauss–Markov Theorem and the Least Square Estimator 304 23.3 The Modified Gauss–Markov Theorem and the Linear Bayes Estimator 306 23.4 Summary 311 Exercises 311 Section 24 Ridge Regression-Type Estimators 314 24.1 Introduction 314 24.2 Minimizing a Second-Degree Form with Respect to a Quadratic Constraint 314 24.3 The Generalized Ridge Regression Estimators 315 24.4 The Mean Square Error of the Generalized Ridge Estimator without Averaging over the Prior Distribution 317 24.5 The Mean Square Error Averaging over the Prior Distribution 321 24.6 Summary 321 Exercises 321 Answers to Selected Exercises 324 References 366 Index 368