{"product_id":"matrix-algebra-for-linear-models-9781118592557","title":"Matrix Algebra for Linear Models","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003eA 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.\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTrade Review\u003c\/b\u003e\u003cbr\u003e\u003cp\u003e“This book seems suitable for an advanced undergraduate and\/or introductory master's level course . . . Four appealing features of this book are its inclusion of an overview, a summary, exercises (with answers provided), and numerical examples for all sections.”  (\u003ci\u003eAmerican Mathematical Society\u003c\/i\u003e, 1 November 2015)\u003c\/p\u003e \u003cp\u003e“The book is suitable for graduate and postgraduate students and researchers. This book is highly recommended.”  (\u003ci\u003eZentralblatt\u003c\/i\u003e, 1 April 2015)\u003c\/p\u003e \u003cp\u003e“This is an excellent and comprehensive presentation of the use of matrices for linear models. The writing is very clear, and the layout is excellent. It would serve well either as a class text or as the foundation for individual personal study.”  (\u003ci\u003eInternational Statistical Review\u003c\/i\u003e, 18 March 2014)\u003c\/p\u003e\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003e\u003cp\u003ePreface xiii\u003c\/p\u003e \u003cp\u003eAcknowledgments xv\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePart I Basic Ideas about Matrices and Systems of Linear Equations 1\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003eSection 1 What Matrices are and Some Basic Operations with Them 3\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1.1 Introduction 3\u003c\/p\u003e \u003cp\u003e1.2 What are Matrices and why are they Interesting to a Statistician? 3\u003c\/p\u003e \u003cp\u003e1.3 Matrix Notation Addition and Multiplication 6\u003c\/p\u003e \u003cp\u003e1.4 Summary 10\u003c\/p\u003e \u003cp\u003eExercises 10\u003c\/p\u003e \u003cp\u003e\u003cb\u003eSection 2 Determinants and Solving a System of Equations 14\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e2.1 Introduction 14\u003c\/p\u003e \u003cp\u003e2.2 Definition of and Formulae for Expanding Determinants 14\u003c\/p\u003e \u003cp\u003e2.3 Some Computational Tricks for the Evaluation of Determinants 16\u003c\/p\u003e \u003cp\u003e2.4 Solution to Linear Equations Using Determinants 18\u003c\/p\u003e \u003cp\u003e2.5 Gauss Elimination 22\u003c\/p\u003e \u003cp\u003e2.6 Summary 27\u003c\/p\u003e \u003cp\u003eExercises 27\u003c\/p\u003e \u003cp\u003e\u003cb\u003eSection 3 The Inverse of a Matrix 30\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e3.1 Introduction 30\u003c\/p\u003e \u003cp\u003e3.2 The Adjoint Method of Finding the Inverse of a Matrix 30\u003c\/p\u003e \u003cp\u003e3.3 Using Elementary Row Operations 31\u003c\/p\u003e \u003cp\u003e3.4 Using the Matrix Inverse to Solve a System of Equations 33\u003c\/p\u003e \u003cp\u003e3.5 Partitioned Matrices and Their Inverses 34\u003c\/p\u003e \u003cp\u003e3.6 Finding the Least Square Estimator 38\u003c\/p\u003e \u003cp\u003e3.7 Summary 44\u003c\/p\u003e \u003cp\u003eExercises 44\u003c\/p\u003e \u003cp\u003e\u003cb\u003eSection 4 Special Matrices and Facts about Matrices that will be used in the Sequel 47\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e4.1 Introduction 47\u003c\/p\u003e \u003cp\u003e4.2 Matrices of the Form aI\u003csub\u003en\u003c\/sub\u003e + bJ\u003csub\u003en\u003c\/sub\u003e 47\u003c\/p\u003e \u003cp\u003e4.3 Orthogonal Matrices 49\u003c\/p\u003e \u003cp\u003e4.4 Direct Product of Matrices 52\u003c\/p\u003e \u003cp\u003e4.5 An Important Property of Determinants 53\u003c\/p\u003e \u003cp\u003e4.6 The Trace of a Matrix 56\u003c\/p\u003e \u003cp\u003e4.7 Matrix Differentiation 57\u003c\/p\u003e \u003cp\u003e4.8 The Least Square Estimator Again 62\u003c\/p\u003e \u003cp\u003e4.9 Summary 62\u003c\/p\u003e \u003cp\u003eExercises 63\u003c\/p\u003e \u003cp\u003e\u003cb\u003eSection 5 Vector Spaces 66\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e5.1 Introduction 66\u003c\/p\u003e \u003cp\u003e5.2 What is a Vector Space? 66\u003c\/p\u003e \u003cp\u003e5.3 The Dimension of a Vector Space 68\u003c\/p\u003e \u003cp\u003e5.4 Inner Product Spaces 70\u003c\/p\u003e \u003cp\u003e5.5 Linear Transformations 73\u003c\/p\u003e \u003cp\u003e5.6 Summary 76\u003c\/p\u003e \u003cp\u003eExercises 76\u003c\/p\u003e \u003cp\u003e\u003cb\u003eSection 6 The Rank of a Matrix and Solutions to Systems of Equations 79\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e6.1 Introduction 79\u003c\/p\u003e \u003cp\u003e6.2 The Rank of a Matrix 79\u003c\/p\u003e \u003cp\u003e6.3 Solving Systems of Equations with Coefficient Matrix of Less than Full Rank 84\u003c\/p\u003e \u003cp\u003e6.4 Summary 87\u003c\/p\u003e \u003cp\u003eExercises 87\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePart II Eigenvalues the Singular Value Decomposition and Principal Components 91\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003eSection 7 Finding the Eigenvalues of a Matrix 93\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e7.1 Introduction 93\u003c\/p\u003e \u003cp\u003e7.2 Eigenvalues and Eigenvectors of a Matrix 93\u003c\/p\u003e \u003cp\u003e7.3 Nonnegative Definite Matrices 101\u003c\/p\u003e \u003cp\u003e7.4 Summary 104\u003c\/p\u003e \u003cp\u003eExercises 105\u003c\/p\u003e \u003cp\u003e\u003cb\u003eSection 8 The Eigenvalues and Eigenvectors of Special Matrices 108\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e8.1 Introduction 108\u003c\/p\u003e \u003cp\u003e8.2 Orthogonal Nonsingular and Idempotent Matrices 109\u003c\/p\u003e \u003cp\u003e8.3 The Cayley–Hamilton Theorem 112\u003c\/p\u003e \u003cp\u003e8.4 The Relationship between the Trace the Determinant and the Eigenvalues of a Matrix 114\u003c\/p\u003e \u003cp\u003e8.5 The Eigenvalues and Eigenvectors of the Kronecker Product of Two Matrices 116\u003c\/p\u003e \u003cp\u003e8.6 The Eigenvalues and the Eigenvectors of a Matrix of the Form aI + bJ 117\u003c\/p\u003e \u003cp\u003e8.7 The Loewner Ordering 119\u003c\/p\u003e \u003cp\u003e8.8 Summary 121\u003c\/p\u003e \u003cp\u003eExercises 122\u003c\/p\u003e \u003cp\u003e\u003cb\u003eSection 9 The Singular Value Decomposition (SVD) 124\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e9.1 Introduction 124\u003c\/p\u003e \u003cp\u003e9.2 The Existence of the SVD 125\u003c\/p\u003e \u003cp\u003e9.3 Uses and Examples of the SVD 127\u003c\/p\u003e \u003cp\u003e9.4 Summary 134\u003c\/p\u003e \u003cp\u003eExercises 134\u003c\/p\u003e \u003cp\u003e\u003cb\u003eSection 10 Applications of the Singular Value Decomposition 137\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e10.1 Introduction 137\u003c\/p\u003e \u003cp\u003e10.2 Reparameterization of a Non-full-Rank Model to a Full-Rank Model 137\u003c\/p\u003e \u003cp\u003e10.3 Principal Components 141\u003c\/p\u003e \u003cp\u003e10.4 The Multicollinearity Problem 143\u003c\/p\u003e \u003cp\u003e10.5 Summary 144\u003c\/p\u003e \u003cp\u003eExercises 145\u003c\/p\u003e \u003cp\u003e\u003cb\u003eSection 11 Relative Eigenvalues and Generalizations of the Singular Value Decomposition 146\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e11.1 Introduction 146\u003c\/p\u003e \u003cp\u003e11.2 Relative Eigenvalues and Eigenvectors 146\u003c\/p\u003e \u003cp\u003e11.3 Generalizations of the Singular Value Decomposition:Overview 151\u003c\/p\u003e \u003cp\u003e11.4 The First Generalization 152\u003c\/p\u003e \u003cp\u003e11.5 The Second Generalization 157\u003c\/p\u003e \u003cp\u003e11.6 Summary 160\u003c\/p\u003e \u003cp\u003eExercises 160\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePart III Generalized Inverses 163\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003eSection 12 Basic Ideas about Generalized Inverses 165\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e12.1 Introduction 165\u003c\/p\u003e \u003cp\u003e12.2 What is a Generalized Inverse and how is One Obtained? 165\u003c\/p\u003e \u003cp\u003e12.3 The Moore–Penrose Inverse 170\u003c\/p\u003e \u003cp\u003e12.4 Summary 173\u003c\/p\u003e \u003cp\u003eExercises 173\u003c\/p\u003e \u003cp\u003e\u003cb\u003eSection 13 Characterizations of Generalized Inverses Using the Singular Value Decomposition 175\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e13.1 Introduction 175\u003c\/p\u003e \u003cp\u003e13.2 Characterization of the Moore–Penrose Inverse 175\u003c\/p\u003e \u003cp\u003e13.3 Generalized Inverses in Terms of the Moore–Penrose Inverse 177\u003c\/p\u003e \u003cp\u003e13.4 Summary 185\u003c\/p\u003e \u003cp\u003eExercises 186\u003c\/p\u003e \u003cp\u003e\u003cb\u003eSection 14 Least Square and Minimum Norm Generalized Inverses 188\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e14.1 Introduction 188\u003c\/p\u003e \u003cp\u003e14.2 Minimum Norm Generalized Inverses 189\u003c\/p\u003e \u003cp\u003e14.3 Least Square Generalized Inverses 193\u003c\/p\u003e \u003cp\u003e14.4 An Extension of Theorem 7.3 to Positive-Semi-definite Matrices 196\u003c\/p\u003e \u003cp\u003e14.5 Summary 197\u003c\/p\u003e \u003cp\u003eExercises 197\u003c\/p\u003e \u003cp\u003e\u003cb\u003eSection 15 More Representations of Generalized Inverses 200\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e15.1 Introduction 200\u003c\/p\u003e \u003cp\u003e15.2 Another Characterization of the Moore–Penrose Inverse 200\u003c\/p\u003e \u003cp\u003e15.3 Still another Representation of the Generalized Inverse 204\u003c\/p\u003e \u003cp\u003e15.4 The Generalized Inverse of a Partitioned Matrix 207\u003c\/p\u003e \u003cp\u003e15.5 Summary 211\u003c\/p\u003e \u003cp\u003eExercises 211\u003c\/p\u003e \u003cp\u003e\u003cb\u003eSection 16 Least Square Estimators for Less than Full-Rank Models 213\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e16.1 Introduction 213\u003c\/p\u003e \u003cp\u003e16.2 Some Preliminaries 213\u003c\/p\u003e \u003cp\u003e16.3 Obtaining the LS Estimator 214\u003c\/p\u003e \u003cp\u003e16.4 Summary 221\u003c\/p\u003e \u003cp\u003eExercises 221\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePart IV Quadratic Forms and the Analysis of Variance 223\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003eSection 17 Quadratic Forms and their Probability Distributions 225\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e17.1 Introduction 225\u003c\/p\u003e \u003cp\u003e17.2 Examples of Quadratic Forms 225\u003c\/p\u003e \u003cp\u003e17.3 The Chi-Square Distribution 228\u003c\/p\u003e \u003cp\u003e17.4 When does the Quadratic Form of a Random Variable have a Chi-Square Distribution? 230\u003c\/p\u003e \u003cp\u003e17.5 When are Two Quadratic Forms with the Chi-Square Distribution Independent? 231\u003c\/p\u003e \u003cp\u003e17.6 Summary 234\u003c\/p\u003e \u003cp\u003eExercises 235\u003c\/p\u003e \u003cp\u003e\u003cb\u003eSection 18 Analysis of Variance: Regression Models and the One- and Two-Way Classification 237\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e18.1 Introduction 237\u003c\/p\u003e \u003cp\u003e18.2 The Full-Rank General Linear Regression Model 237\u003c\/p\u003e \u003cp\u003e18.3 Analysis of Variance: One-Way Classification 241\u003c\/p\u003e \u003cp\u003e18.4 Analysis of Variance: Two-Way Classification 244\u003c\/p\u003e \u003cp\u003e18.5 Summary 249\u003c\/p\u003e \u003cp\u003eExercises 249\u003c\/p\u003e \u003cp\u003e\u003cb\u003eSection 19 More ANOVA 253\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e19.1 Introduction 253\u003c\/p\u003e \u003cp\u003e19.2 The Two-Way Classification with Interaction 254\u003c\/p\u003e \u003cp\u003e19.3 The Two-Way Classification with One Factor Nested 258\u003c\/p\u003e \u003cp\u003e19.4 Summary 262\u003c\/p\u003e \u003cp\u003eExercises 262\u003c\/p\u003e \u003cp\u003e\u003cb\u003eSection 20 The General Linear Hypothesis 264\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e20.1 Introduction 264\u003c\/p\u003e \u003cp\u003e20.2 The Full-Rank Case 264\u003c\/p\u003e \u003cp\u003e20.3 The Non-full-Rank Case 267\u003c\/p\u003e \u003cp\u003e20.4 Contrasts 270\u003c\/p\u003e \u003cp\u003e20.5 Summary 273\u003c\/p\u003e \u003cp\u003eExercises 273\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePart V Matrix Optimization Problems 275\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003eSection 21 Unconstrained Optimization Problems 277\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e21.1 Introduction 277\u003c\/p\u003e \u003cp\u003e21.2 Unconstrained Optimization Problems 277\u003c\/p\u003e \u003cp\u003e21.3 The Least Square Estimator Again 281\u003c\/p\u003e \u003cp\u003e21.4 Summary 283\u003c\/p\u003e \u003cp\u003eExercises 283\u003c\/p\u003e \u003cp\u003e\u003cb\u003eSection 22 Constrained Minimization Problems with Linear Constraints 287\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e22.1 Introduction 287\u003c\/p\u003e \u003cp\u003e22.2 An Overview of Lagrange Multipliers 287\u003c\/p\u003e \u003cp\u003e22.3 Minimizing a Second-Degree Form with Respect to a Linear Constraint 293\u003c\/p\u003e \u003cp\u003e22.4 The Constrained Least Square Estimator 295\u003c\/p\u003e \u003cp\u003e22.5 Canonical Correlation 299\u003c\/p\u003e \u003cp\u003e22.6 Summary 302\u003c\/p\u003e \u003cp\u003eExercises 302\u003c\/p\u003e \u003cp\u003e\u003cb\u003eSection 23 The Gauss–Markov Theorem 304\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e23.1 Introduction 304\u003c\/p\u003e \u003cp\u003e23.2 The Gauss–Markov Theorem and the Least Square Estimator 304\u003c\/p\u003e \u003cp\u003e23.3 The Modified Gauss–Markov Theorem and the Linear Bayes Estimator 306\u003c\/p\u003e \u003cp\u003e23.4 Summary 311\u003c\/p\u003e \u003cp\u003eExercises 311\u003c\/p\u003e \u003cp\u003e\u003cb\u003eSection 24 Ridge Regression-Type Estimators 314\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e24.1 Introduction 314\u003c\/p\u003e \u003cp\u003e24.2 Minimizing a Second-Degree Form with Respect to a Quadratic Constraint 314\u003c\/p\u003e \u003cp\u003e24.3 The Generalized Ridge Regression Estimators 315\u003c\/p\u003e \u003cp\u003e24.4 The Mean Square Error of the Generalized Ridge Estimator without Averaging over the Prior Distribution 317\u003c\/p\u003e \u003cp\u003e24.5 The Mean Square Error Averaging over the Prior Distribution 321\u003c\/p\u003e \u003cp\u003e24.6 Summary 321\u003c\/p\u003e \u003cp\u003eExercises 321\u003c\/p\u003e \u003cp\u003eAnswers to Selected Exercises 324\u003c\/p\u003e \u003cp\u003eReferences 366\u003c\/p\u003e \u003cp\u003eIndex 368\u003c\/p\u003e","brand":"John Wiley \u0026 Sons Inc","offers":[{"title":"Default Title","offer_id":49406893064535,"sku":"9781118592557","price":91.76,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9781118592557.jpg?v=1730497468","url":"https:\/\/bookcurl.com\/products\/matrix-algebra-for-linear-models-9781118592557","provider":"Book Curl","version":"1.0","type":"link"}