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A thoroughly updated guide to matrix algebra and it uses in statistical analysis and features SAS®, MATLAB®, and R throughout
This Second Edition addresses matrix algebra that is useful in the statistical analysis of data as well as within statistics as a whole. The material is presented in an explanatory style rather than a formal theorem-proof format and is self-contained. Featuring numerous applied illustrations, numerical examples, and exercises, the book has been updated to include the use of SAS, MATLAB, and R for the execution of matrix computations. In addition, André I. Khuri, who has extensive research and teaching experience in the field, joins this new edition as co-author. The Second Edition also:
Matrix Algebra Useful for Statistics, Second Edition is an ideal textbook for advanced undergraduate and first-year graduate level courses in statistics and other related disciplines. The book is also appropriate as a reference for independent readers who use statistics and wish to improve their knowledge of matrix algebra.
THE LATE SHAYLE R. SEARLE, PHD, was professor emeritus of biometry at Cornell University. He was the author of Linear Models for Unbalanced Data and Linear Models and co-author of Generalized, Linear, and Mixed Models, Second Edition, Matrix Algebra for Applied Economics, and Variance Components, all published by Wiley. Dr. Searle received the Alexander von Humboldt Senior Scientist Award, and he was an honorary fellow of the Royal Society of New Zealand.
ANDRÉ I. KHURI, PHD, is Professor Emeritus of Statistics at the University of Florida. He is the author of Advanced Calculus with Applications in Statistics, Second Edition and co-author of Statistical Tests for Mixed Linear Models, all published by Wiley. Dr. Khuri is a member of numerous academic associations, among them the American Statistical Association and the Institute of Mathematical Statistics.
The late Shayle R. Searle, PhD, was professor emeritus of biometry at Cornell University. He was the author of Linear Models for Unbalanced Data and Linear Models and co-author of Generalized, Linear, and Mixed Models, Second Edition, Matrix Algebra for Applied Economics, and Variance Components, all published by Wiley. Dr. Searle received the Alexander von Humboldt Senior Scientist Award, and he was an honorary fellow of the Royal Society of New Zealand.
André I. Khuri, PhD, is Professor Emeritus of Statistics at the University of Florida. He is the author of Advanced Calculus with Applications in Statistics, Second Edition and co-author of Statistical Tests for Mixed Linear Models, all published by Wiley. Dr. Khuri is a member of numerous academic associations, among them the American Statistical Association and the Institute of Mathematical Statistics.
PREFACE xvii
PREFACE TO THE FIRST EDITION xix
INTRODUCTION xxi
ABOUT THE COMPANION WEBSITE xxxi
PART I DEFINITIONS, BASIC CONCEPTS, AND MATRIX OPERATIONS 1
1 Vector Spaces, Subspaces, and Linear Transformations 3
1.1 Vector Spaces 3
1.2 Base of a Vector Space 5
1.3 Linear Transformations 7
2 Matrix Notation and Terminology 11
2.1 Plotting of a Matrix 14
2.2 Vectors and Scalars 16
2.3 General Notation 16
3 Determinants 21
3.1 Expansion by Minors 21
3.2 Formal Definition 25
3.3 Basic Properties 27
3.4 Elementary Row Operations 34
3.5 Examples 37
3.6 Diagonal Expansion 39
3.7 The Laplace Expansion 42
3.8 Sums and Differences of Determinants 44
3.9 A Graphical Representation of a 3 × 3 Determinant 45
4 Matrix Operations 51
4.1 The Transpose of a Matrix 51
4.2 Partitioned Matrices 52
4.3 The Trace of a Matrix 55
4.4 Addition 56
4.5 Scalar Multiplication 58
4.6 Equality and the Null Matrix 58
4.7 Multiplication 59
4.8 The Laws of Algebra 74
4.9 Contrasts With Scalar Algebra 76
4.10 Direct Sum of Matrices 77
4.11 Direct Product of Matrices 78
4.12 The Inverse of a Matrix 80
4.13 Rank of a Matrix—Some Preliminary Results 82
4.14 The Number of LIN Rows and Columns in a Matrix 84
4.15 Determination of the Rank of a Matrix 85
4.16 Rank and Inverse Matrices 87
4.17 Permutation Matrices 87
5 Special Matrices 97
5.1 Symmetric Matrices 97
5.2 Matrices Having All Elements Equal 102
5.3 Idempotent Matrices 104
5.4 Orthogonal Matrices 106
5.5 Parameterization of Orthogonal Matrices 109
5.6 Quadratic Forms 110
5.7 Positive Definite Matrices 113
6 Eigenvalues and Eigenvectors 119
6.1 Derivation of Eigenvalues 119
6.2 Elementary Properties of Eigenvalues 122
6.3 Calculating Eigenvectors 125
6.4 The Similar Canonical Form 128
6.5 Symmetric Matrices 131
6.6 Eigenvalues of Orthogonal and Idempotent Matrices 135
6.7 Eigenvalues of Direct Products and Direct Sums of Matrices 138
6.8 Nonzero Eigenvalues of AB and BA 140
7 Diagonalization of Matrices 145
7.1 Proving the Diagonability Theorem 145
7.2 Other Results for Symmetric Matrices 148
7.3 The Cayley–Hamilton Theorem 152
7.4 The Singular-Value Decomposition 153
8 Generalized Inverses 159
8.1 The Moore–Penrose Inverse 159
8.2 Generalized Inverses 160
8.3 Other Names and Symbols 164
8.4 Symmetric Matrices 165
9 Matrix Calculus 171
9.1 Matrix Functions 171
9.2 Iterative Solution of Nonlinear Equations 174
9.3 Vectors of Differential Operators 175
9.4 Vec and Vech Operators 179
9.5 Other Calculus Results 181
9.6 Matrices with Elements That Are Complex Numbers 188
9.7 Matrix Inequalities 189
PART II APPLICATIONS OF MATRICES IN STATISTICS 199
10 Multivariate Distributions and Quadratic Forms 201
10.1 Variance-Covariance Matrices 202
10.2 Correlation Matrices 203
10.3 Matrices of Sums of Squares and Cross-Products 204
10.4 The Multivariate Normal Distribution 207
10.5 Quadratic Forms and ;;2-Distributions 208
10.6 Computing the Cumulative Distribution Function of a Quadratic Form 213
11 Matrix Algebra of Full-Rank Linear Models 219
11.1 Estimation of ;; by the Method of Least Squares 220
11.2 Statistical Properties of the Least-Squares Estimator 226
11.3 Multiple Correlation Coefficient 229
11.4 Statistical Properties under the Normality Assumption 231
11.5 Analysis of Variance 233
11.6 The Gauss–Markov Theorem 234
11.7 Testing Linear Hypotheses 237
11.8 Fitting Subsets of the x-Variables 246
11.9 The Use of the R(.|.) Notation in Hypothesis Testing 247
12 Less-Than-Full-Rank Linear Models 253
12.1 General Description 253
12.2 The Normal Equations 256
12.3 Solving the Normal Equations 257
12.4 Expected Values and Variances 259
12.5 Predicted y-Values 260
12.6 Estimating the Error Variance 261
12.7 Partitioning the Total Sum of Squares 262
12.8 Analysis of Variance 263
12.9 The R(⋅|⋅) Notation 265
12.10 Estimable Linear Functions 266
12.11 Confidence Intervals 272
12.12 Some Particular Models 272
12.13 The R(⋅|⋅) Notation (Continued) 277
12.14 Reparameterization to a Full-Rank Model 281
13 Analysis of Balanced Linear Models Using Direct Products of Matrices 287
13.1 General Notation for Balanced Linear Models 289
13.2 Properties Associated with Balanced Linear Models 293
13.3 Analysis of Balanced Linear Models 298
14 Multiresponse Models 313
14.1 Multiresponse Estimation of Parameters 314
14.2 Linear Multiresponse Models 316
14.3 Lack of Fit of a Linear Multiresponse Model 318
PART III MATRIX COMPUTATIONS AND RELATED SOFTWARE 327
15 SAS/IML 329
15.1 Getting Started 329
15.2 Defining a Matrix 329
15.3 Creating a Matrix 330
15.4 Matrix Operations 331
15.5 Explanations of SAS Statements Used Earlier in the Text 354
16 Use of MATLAB in Matrix Computations 363
16.1 Arithmetic Operators 363
16.2 Mathematical Functions 364
16.3 Construction of Matrices 365
16.4 Two- and Three-Dimensional Plots 371
17 Use of R in Matrix Computations 383
17.1 Two- and Three-Dimensional Plots 396
Exercises 408
APPENDIX 413
INDEX 475
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