Matrix Analysis for Statistics

Preface xi About the Companion Website xv 1 A Review of Elementary Matrix Algebra 1 1.1 Introduction 1 1.2 Definitions and Notation 1 1.3 Matrix Addition and Multiplication 2 1.4 The Transpose 3 1.5 The Trace 4 1.6 The Determinant 5 1.7 The Inverse 9 1.8 Partitioned Matrices 12 1.9 The Rank of a Matrix 14 1.10 Orthogonal Matrices 15 1.11 Quadratic Forms 16 1.12 Complex Matrices 18 1.13 Random Vectors and Some Related Statistical Concepts 19 Problems 29 2 Vector Spaces 35 2.1 Introduction 35 2.2 Definitions 35 2.3 Linear Independence and Dependence 42 2.4 Matrix Rank and Linear Independence 45 2.5 Bases and Dimension 49 2.6 Orthonormal Bases and Projections 53 2.7 Projection Matrices 58 2.8 Linear Transformations and Systems of Linear Equations 65 2.9 The Intersection and Sum of Vector Spaces 73 2.10 Oblique Projections 76 2.11 Convex Sets 80 Problems 85 3 Eigenvalues and Eigenvectors 95 3.1 Introduction 95 3.2 Eigenvalues, Eigenvectors, and Eigenspaces 95 3.3 Some Basic Properties of Eigenvalues and Eigenvectors 99 3.4 Symmetric Matrices 106 3.5 Continuity of Eigenvalues and Eigenprojections 114 3.6 Extremal Properties of Eigenvalues 116 3.7 Additional Results Concerning Eigenvalues Of Symmetric Matrices 123 3.8 Nonnegative Definite Matrices 129 3.9 Antieigenvalues and Antieigenvectors 141 Problems 144 4 Matrix Factorizations and Matrix Norms 155 4.1 Introduction 155 4.2 The Singular Value Decomposition 155 4.3 The Spectral Decomposition of a Symmetric Matrix 162 4.4 The Diagonalization of a Square Matrix 169 4.5 The Jordan Decomposition 173 4.6 The Schur Decomposition 175 4.7 The Simultaneous Diagonalization of Two Symmetric Matrices 178 4.8 Matrix Norms 184 Problems 191 5 Generalized Inverses 201 5.1 Introduction 201 5.2 The Moore-Penrose Generalized Inverse 202 5.3 Some Basic Properties of the Moore-Penrose Inverse 205 5.4 The Moore-Penrose Inverse of a Matrix Product 211 5.5 The Moore-Penrose Inverse of Partitioned Matrices 215 5.6 The Moore-Penrose Inverse of a Sum 219 5.7 The Continuity of the Moore-Penrose Inverse 222 5.8 Some Other Generalized Inverses 224 5.9 Computing Generalized Inverses 232 Problems 238 6 Systems of Linear Equations 247 6.1 Introduction 247 6.2 Consistency of a System of Equations 247 6.3 Solutions to a Consistent System of Equations 251 6.4 Homogeneous Systems of Equations 258 6.5 Least Squares Solutions to a System of Linear Equations 260 6.6 Least Squares Estimation For Less Than Full Rank Models 266 6.7 Systems of Linear Equations and The Singular Value Decomposition 271 6.8 Sparse Linear Systems of Equations 273 Problems 278 7 Partitioned Matrices 285 7.1 Introduction 285 7.2 The Inverse 285 7.3 The Determinant 288 7.4 Rank 296 7.5 Generalized Inverses 298 7.6 Eigenvalues 302 Problems 307 8 Special Matrices and Matrix Operations 315 8.1 Introduction 315 8.2 The Kronecker Product 315 8.3 The Direct Sum 323 8.4 The Vec Operator 323 8.5 The Hadamard Product 329 8.6 The Commutation Matrix 339 8.7 Some Other Matrices Associated With the Vec Operator 346 8.8 Nonnegative Matrices 351 8.9 Circulant and Toeplitz Matrices 363 8.10 Hadamard and Vandermonde Matrices 369 Problems 373 9 Matrix Derivatives and Related Topics 387 9.1 Introduction 387 9.2 Multivariable Differential Calculus 387 9.3 Vector and Matrix Functions 390 9.4 Some Useful Matrix Derivatives 396 9.5 Derivatives of Functions of Patterned Matrices 400 9.6 The Perturbation Method 402 9.7 Maxima and Minima 409 9.8 Convex and Concave Functions 413 9.9 The Method of Lagrange Multipliers 417 Problems 423 10 Inequalities 433 10.1 Introduction 433 10.2 Majorization 433 10.3 Cauchy-Schwarz Inequalities 444 10.4 H¨older's Inequality 446 10.5 Minkowski's Inequality 450 10.6 The Arithmetic-Geometric Mean Inequality 452 Problems 453 11 Some Special Topics Related to Quadratic Forms 457 11.1 Introduction 457 11.2 Some Results on Idempotent Matrices 457 11.3 Cochran's Theorem 462 11.4 Distribution of Quadratic Forms in Normal Variates 465 11.5 Independence of Quadratic Forms 471 11.6 Expected Values of Quadratic Forms 477 11.7 The Wishart Distribution 485 Problems 496 References 507 Index 513