Table of contents for A matrix handbook for statisticians / George A.F. Seber.

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CONTENTS
Preface iv
1 Notation 1
1.1 General Definitions 1
1.2 Some Continuous Univariate Distributions 2
1.3 Glossary of Notation 3
2 Vectors, Vector Spaces, and Convexity 7
2.1 Vector Spaces 7
2.1.1 Definitions 7
2.1.2 Quadratic Subspaces 9
2.1.3 Sums and Intersections of Subspaces 10
2.1.4 Span and Basis 11
2.1.5 Isomorphism 12
2.2 Inner Products 13
2.2.1 Definition and Properties 13
2.2.2 Functionals 15
2.2.3 Orthogonality 16
2.2.4 Column and Null Spaces 18
2.3 Projections 20
2.3.1 General Projections 20
2.3.2 Orthogonal Projections 21
2.4 Metric Spaces 26
2.5 Convex Sets and Functions 27
2.6 Coordinate Geometry 31
2.6.1 Hyperplanes and Lines 31
2.6.2 Quadratics 31
2.6.3 Miscellaneous Results 32
3 Rank 33
3.1 Some General Properties 33
3.2 Matrix Products 35
3.3 Matrix Cancellation Rules 37
3.4 Matrix Sums 38
3.5 Matrix Differences 42
3.6 Partitioned Matrices 44
3.7 Maximal and Minimal Ranks 47
3.8 Matrix Index 49
4 Matrix Functions: Inverse, Transpose, Trace, Determinant, and Norm 51
4.1 Inverse 51
4.2 Transpose 52
4.3 Trace 52
4.4 Determinants 55
4.4.1 Introduction 55
4.4.2 Adjoint Matrix 57
4.4.3 Compound Matrix 59
4.4.4 Expansion of a Determinant 59
4.5 Permanents 61
4.6 Norms 63
4.6.1 Vector Norms 63
4.6.2 Matrix Norms 65
4.6.3 Unitarily Invariant Norms 71
4.6.4 M,N-Invariant Norms 75
4.6.5 Computational Accuracy 75
5 Complex, Hermitian, and Related Matrices 77
5.1 Complex Matrices 77
5.1.1 Some General Results 78
5.1.2 Determinants 79
5.2 Hermitian Matrices 80
5.3 Skew-Hermitian Matrices 81
5.4 Complex Symmetric Matrices 82
5.5 Real Skew-Symmetric Matrices 83
5.6 Normal Matrices 84
5.7 Quaternions 85
6 Eigenvalues, Eigenvectors, and Singular Values 87
6.1 Introduction and Definitions 87
6.1.1 Characteristic Polynomial 88
6.1.2 Eigenvalues 91
6.1.3 Singular Values 97
6.1.4 Functions of a Matrix 99
6.1.5 Eigenvectors 99
6.1.6 Hermitian Matrices 100
6.1.7 Computational Methods 101
6.1.8 Generalized Eigenvalues 102
6.1.9 Matrix Products 103
6.2 Variational Characteristics for Hermitian Matrices 104
6.3 Separation Theorems 107
6.4 Inequalities for Matrix Sums 112
6.5 Inequalities for Matrix Differences 115
6.6 Inequalities for Matrix Products 115
6.7 Antieigenvalues and Antieigenvectors 117
7 Generalized Inverses 121
7.1 Definitions 121
7.2 Weak Inverses 122
7.2.1 General Properties 122
7.2.2 Products 126
7.2.3 Sums and Differences 128
7.2.4 Real Symmetric Matrices 128
7.2.5 Decomposition Methods 129
7.3 Other Inverses 130
7.3.1 Reflexive (g12) Inverse 130
7.3.2 Minimum Norm (g14) Inverse 131
7.3.3 Minimum Norm Reflexive (g124) Inverse 132
7.3.4 Least Squares (g13) Inverse 132
7.3.5 Least Squares Reflexive (g123) Inverse 133
7.4 Moore-Penrose (g1234) Inverse 133
7.4.1 General Properties 133
7.4.2 Sums 139
7.4.3 Products 139
7.5 Group Inverse 141
7.6 Some General Properties of Inverses 141
8 Some Special Matrices 143
8.1 Orthogonal and Unitary Matrices 143
8.2 Permutation Matrices 146
8.3 Circulant, Toeplitz, and Related Matrices 147
8.3.1 Regular Circulant 147
8.3.2 Symmetric Regular Circulant 151
8.3.3 Symmetric Circulant 152
8.3.4 Toeplitz Matrix 153
8.3.5 Persymmetric Matrix 155
8.3.6 Cross-Symmetric (Centrosymmetric) Matrix 155
8.3.7 Block Circulant 156
8.3.8 Hankel Matrix 157
8.4 Diagonally Dominant Matrices 158
8.5 Hadamard Matrices 160
8.6 Idempotent Matrices 161
8.6.1 General Properties 161
8.6.2 Sums of Idempotent Matrices and Extensions 166
8.6.3 Products of Idempotent Matrices 170
8.7 Tripotent Matrices 170
8.8 Irreducible Matrices 172
8.9 Triangular Matrices 173
8.10 Hessenberg Matrices 175
8.11 Tridiagonal Matrices 175
8.12 Vandermonde and Fourier Matrices 178
8.12.1 Vandermonde Matrix 178
8.12.2 Fourier Matrix 180
8.13 Zero-One (0,1) Matrices 181
8.14 Some Miscellaneous Matrices and Arrays 183
8.14.1 Krylov Matrix 183
8.14.2 Nilpotent and Unipotent Matrices 183
8.14.3 Payoff Matrix 184
8.14.4 Stable and Positive Stable Matrices 185
8.14.5 P-Matrix 186
8.14.6 Z- and M-Matrices 186
8.14.7 Three-Dimensional Arrays 188
9 Non-Negative Vectors and Matrices 191
9.1 Introduction 191
9.1.1 Scaling 192
9.1.2 Modulus of a Matrix 193
9.2 Spectral Radius 193
9.2.1 General Properties 193
9.2.2 Dominant Eigenvalue 195
9.3 Canonical Form of a Non-negative Matrix 197
9.4 Irreducible Matrices 198
9.4.1 Irreducible Non-negative Matrix 198
9.4.2 Periodicity 203
9.4.3 Non-negative and Non-positive Off-Diagonal Elements 204
9.4.4 Perron Matrix 205
9.4.5 Decomposable Matrix 206
9.5 Leslie Matrix 206
9.6 Stochastic Matrices 208
9.6.1 Basic Properties 208
9.6.2 Finite Homogeneous Markov Chain 209
9.6.3 Countably Infinite Stochastic Matrix 211
9.6.4 Infinite Irreducible Stochastic Matrix 211
9.7 Doubly Stochastic Matrices 212
10 Positive Definite and Non-negative Definite Matrices 215
10.1 Introduction 215
10.2 Non-negative Definite Matrices 216
10.2.1 Some General Properties 216
10.2.2 Gram Matrix 219
10.2.3 Doubly Non-negative Matrix 219
10.3 Positive Definite Matrices 220
10.4 Pairs of Matrices 223
10.4.1 Non-Negative or Positive Definite Difference 224
10.4.2 One or More Non-Negative Definite Matrices 226
11 Special Products and Operators 229
11.1 Kronecker Product 229
11.1.1 Two Matrices 229
11.1.2 More Than Two Matrices 233
11.2 Vec Operator 235
11.3 Vec-Permutation (Commutation) Matrix 237
11.4 Generalized Vec-Permutation Matrix 241
11.5 Vech Operator 242
11.5.1 Symmetric Matrix 242
11.5.2 Lower Triangular Matrix 246
11.6 Star Operator 247
11.7 Hadamard Product 247
11.8 Rao-Khatri Product 251
12 Inequalities 253
12.1 Cauchy-Schwarz inequalities 253
12.1.1 Real Vector Inequalities and Extensions 254
12.1.2 Complex Vector Inequalities 257
12.1.3 Real Matrix Inequalities 258
12.1.4 Complex Matrix Inequalities 261
12.2 H?older?s Inequality and Extensions 263
12.3 Minkowski?s Inequality and Extensions 264
12.4 Weighted Means 266
12.5 Quasilinearization (Representation Theorems) 267
12.6 Some Geometrical Properties 268
12.7 Miscellaneous Inequalities 269
12.7.1 Determinants 269
12.7.2 Trace 270
12.7.3 Quadratics 271
12.7.4 Sums and Products 271
12.8 Some Identities 273
13 Linear Equations 275
13.1 Unknown vector 275
13.1.1 Consistency 275
13.1.2 Solutions 276
13.1.3 Homogeneous Equations 277
13.1.4 Restricted Equations 278
13.2 Unknown Matrix 278
13.2.1 Consistency 279
13.2.2 Some Special Cases 279
14 Partitioned Matrices 285
14.1 Schur Complement 285
14.2 Inverses 288
14.3 Determinants 292
14.4 Positive and Non-Negative Definite matrices 294
14.5 Eigenvalues 296
14.6 Generalized Inverses 297
14.6.1 Weak Inverses 297
14.6.2 Moore-Penrose Inverses 300
14.7 Miscellaneous partitions 302
15 Patterned Matrices 303
15.1 Inverses 303
15.2 Determinants 307
15.3 Perturbations 308
15.4 Matrices With Repeated Elements and Blocks 312
15.5 Generalized Inverses 315
15.5.1 Weak Inverses 315
15.5.2 Moore-Penrose Inverses 316
16 Factorization of Matrices 319
16.1 Similarity Reductions 319
16.2 Reduction by Elementary Transformations 325
16.2.1 Types of Transformation 325
16.2.2 Equivalence Relation 326
16.2.3 Echelon Form 326
16.2.4 Hermite Form 328
16.3 Singular Value Decomposition (SVD) 330
16.4 Triangular Factorizations 332
16.5 Orthogonal-Triangular Reductions 336
16.6 Further Diagonal or Tridiagonal Reductions 338
16.7 Congruence 341
16.8 Simultaneous Reductions 341
16.9 Polar Decomposition 344
16.10 Miscellaneous Factorizations 344
17 Differentiation and Finite Differences 347
17.1 Introduction 347
17.2 Scalar Differentiation 348
17.2.1 Differentiation with Respect to t 348
17.2.2 Differentiation With Respect to a Vector Element 349
17.2.3 Differentiation With Respect to a Matrix Element 351
17.3 Vector Differentiation: Scalar Function 354
17.3.1 Basic Results 354
17.3.2 x=vec X 355
17.3.3 Function of a Function 356
17.4 Vector Differentiation: Vector Function 357
17.5 Matrix Differentiation: Scalar Function 361
17.5.1 General Results 361
17.5.2 f = trace 362
17.5.3 f = determinant 364
17.5.4 f = yrs 365
17.5.5 f = eigenvalue 366
17.6 Transformation Rules 366
17.7 Matrix Differentiation: Matrix Function 367
17.8 Matrix Differentials 368
17.9 Perturbation Using Differentials 372
17.10 Matrix Linear Differential Equations 373
17.11 Second Order Derivatives 374
17.12 Vector Difference Equations 377
18 Jacobians 379
18.1 Introduction 379
18.2 Method of Differentials 381
18.3 Further Techniques 381
18.3.1 Chain Rule 382
18.3.2 Exterior (Wedge) Product of Differentials 382
18.3.3 Induced Functional Equations 383
18.3.4 Jacobians Involving Transposes 384
18.3.5 Patterned Matrices and L-Structures 384
18.4 Vector Transformations 386
18.5 Jacobians for Complex Vectors and Matrices 387
18.6 Matrices with Functionally Independent Elements 388
18.7 Symmetric and Hermitian Matrices 390
18.8 Skew-Symmetric and Skew-Hermitian Matrices 393
18.9 Triangular Matrices 395
18.9.1 Linear Transformations 395
18.9.2 Nonlinear Transformations of X 397
18.9.3 Decompositions With One matrix Skew Symmetric 399
18.9.4 Symmetric Y 400
18.9.5 Positive Definite Y 401
18.9.6 Hermitian Positive Definite Y 402
18.9.7 Skew Symmetric Y 402
18.9.8 LU Decomposition 403
18.10 Decompositions Involving Diagonal Matrices 403
18.10.1 Square Matrices 403
18.10.2 One Triangular Matrix 404
18.10.3 Symmetric and Skew Symmetric Matrices 407
18.11 Positive?Definite Matrices 407
18.12 Caley Transformation 408
18.13 Diagonalizable Matrices 410
18.14 Pairs of Matrices 411
19 Matrix Limits, Sequences and Series 413
19.1 Limits 413
19.2 Sequences 414
19.3 Asymptotically Equivalent Sequences 416
19.4 Series 417
19.5 Matrix Functions 418
19.6 Matrix Exponentials 419
20 Random Vectors 423
20.1 Notation 423
20.2 Variances and Covariances 423
20.3 Correlations 426
20.3.1 Population Correlations 426
20.3.2 Sample Correlations 428
20.4 Quadratics 429
20.5 Multivariate Normal Distribution 431
20.5.1 Definition and Properties 431
20.5.2 Quadratics in Normal Variables 434
20.5.3 Quadratics and Chi-squared 438
20.5.4 Independence and Quadratics 438
20.5.5 Independence of Several Quadratics 440
20.6 Complex Random Vectors 441
20.7 Regression Models 442
20.7.1 V is the Identity Matrix 444
20.7.2 V is Positive Definite 449
20.7.3 V is Non-negative Definite 450
20.8 Other Multivariate Distributions 453
20.8.1 Multivariate t-Distribution 453
20.8.2 Elliptical and Spherical Distributions 454
20.8.3 Dirichlet Distributions 456
21 Random Matrices 457
21.1 Introduction 457
21.2 Generalized Quadratic Forms 458
21.2.1 General Results 458
21.2.2 Wishart Distribution 461
21.3 Random Samples 466
21.3.1 One Sample 466
21.3.2 Two Samples 469
21.4 Multivariate Linear Model 470
21.4.1 Least Squares Estimation 470
21.4.2 Statistical Inference 472
21.4.3 Two Extensions 473
21.5 Dimension Reduction Techniques 474
21.5.1 Principal Component Analysis (PCA) 474
21.5.2 Discriminant Coordinates 478
21.5.3 Canonical Correlations and Variates 479
21.5.4 Latent Variable Methods 481
21.5.5 Classical (Metric) Scaling 482
21.6 Procrustes Analysis (Matching Configurations) 483
21.7 Some Specific Random Matrices 484
21.8 Allocation Problems 485
21.9 Matrix Variate Distributions 485
21.10 Matrix Ensembles 488
22 Inequalities for Probabilities and Random Variables 491
22.1 General Probabilities 491
22.2 Bonferroni-Type Inequalities 493
22.3 Distribution-Free Probability Inequalities 494
22.3.1 Chebyshev-Type Inequalities 494
22.3.2 Kolmogorov-Type Inequalities 496
22.3.3 Quadratics and Inequalities 496
22.4 Data Inequalities 497
22.5 Inequalities for Expectations 498
22.6 Multivariate Inequalities 498
22.6.1 Convex Subsets 498
22.6.2 Multivariate Normal 499
22.6.3 Inequalities For Other Distributions 502
23 Majorization 503
23.1 General Properties 503
23.2 Schur Convexity 507
23.3 Probabilities and Random variables 509
24 Optimization and Matrix Approximation 511
24.1 Stationary Values 511
24.2 Using Convex and Concave Functions 513
24.3 Two General Methods 514
24.3.1 Maximum Likelihood 514
24.3.2 Least Squares 516
24.4 Optimizing a Function of a Matrix 516
24.4.1 Trace 516
24.4.2 Norm 518
24.4.3 Quadratics 521
24.5 Optimal Designs 523
References 524
Index 541

Library of Congress Subject Headings for this publication:

Matrices.
Statistics.