Generalized Inverses: Theory and Applications

Contents......Page 10
Preface to the Second Edition......Page 6
From the Preface to the First Edition......Page 8
C......Page 14
S......Page 15
Z......Page 16
2. Generalized Inverses of Matrices......Page 17
3. Illustration: Solvability of Linear Systems......Page 18
4. Diversity of Generalized Inverses......Page 19
6. Historical Note......Page 20
Suggested Further Reading......Page 21
1. Scalars and Vectors......Page 22
2. Linear Transformations and Matrices......Page 26
3. Elementary Operations and Permutations......Page 38
4. The Hermite Normal Form and Related Items......Page 39
5. Determinants and Volume......Page 44
6. Some Multilinear Algebra......Page 48
7. The Jordan Normal Form......Page 50
8. The Smith Normal Form......Page 54
Suggested Further Reading......Page 55
1. The Penrose Equations......Page 56
2. Existence and Construction of {1}-Inverses......Page 57
3. Properties of {1}-Inverses......Page 58
4. Existence and Construction of {1, 2}-Inverses......Page 61
5. Existence and Construction of {1, 2, 3}-, {1, 2, 4}-, and {1, 2, 3, 4}-Inverses......Page 62
6. Explicit Formula for A†......Page 64
7. Construction of {2}-Inverses of Prescribed Rank......Page 65
Suggested Further Reading......Page 67
1. Solutions of Linear Systems......Page 68
2. Characterization of A{1, 3} and A{1, 4}......Page 71
3. Characterization of A{2}, A{1, 2}, and Other Subsets of A{2}......Page 72
4. Idempotent Matrices and Projectors......Page 74
5. Matrix Functions......Page 81
6. Generalized Inverses with Prescribed Range and Null Space......Page 87
7. Orthogonal Projections and Orthogonal Projectors......Page 90
8. Efficient Characterization of Classes of Generalized Inverses......Page 101
9. Restricted Generalized Inverses......Page 104
10. The Bott–Duffin Inverse......Page 107
11. An Application of {1}-Inverses in Interval Linear Programming......Page 111
12. A {1, 2}-Inverse for the Integral Solution of Linear Equations......Page 113
13. An Application of the Bott–Duffin Inverse to Electrical Networks......Page 115
Suggested Further Reading......Page 119
1. Least-Squares Solutions of Inconsistent Linear Systems......Page 120
2. Solutions of Minimum Norm......Page 124
3. Tikhonov Regularization......Page 130
4. Weighted Generalized Inverses......Page 133
5. Least-Squares Solutions and Basic Solutions......Page 138
6. Minors of the Moore–Penrose Inverse......Page 143
7. Essentially Strictly Convex Norms and the Associated Projectors and Generalized Inverses......Page 146
8. An Extremal Property of the Bott–Duffin Inverse with Application to Electrical Networks......Page 165
Suggested Further Reading......Page 167
1. Introduction......Page 168
2. The Matrix Index......Page 169
3. Spectral Inverse of a Diagonable Matrix......Page 171
4. The Group Inverse......Page 172
5. Spectral Properties of the Group Inverse......Page 177
6. The Drazin Inverse......Page 179
7. Spectral Properties of the Drazin Inverse......Page 184
8. Index 1-Nilpotent Decomposition of a Square Matrix......Page 185
9. Quasi-Commuting Inverses......Page 187
10. Other Spectral Generalized Inverses......Page 188
Suggested Further Reading......Page 190
2. Partitioned Matrices and Linear Equations......Page 191
3. Intersection of Manifolds......Page 198
4. Common Solutions of Linear Equations and Generalized Inverses of Partitioned Matrices......Page 205
5. Generalized Inverses of Bordered Matrices......Page 212
Suggested Further Reading......Page 216
1. Introduction......Page 217
2. The Singular Value Decomposition......Page 221
3. The Schmidt Approximation Theorem......Page 228
4. Partial Isometries and the Polar Decomposition Theorem......Page 234
5. Principal Angles Between Subspaces......Page 246
6. Perturbations......Page 254
7. A Spectral Theory for Rectangular Matrices......Page 258
8. Generalized Singular Value Decompositions......Page 267
Suggested Further Reading......Page 271
1. Introduction......Page 273
2. Computation of Unrestricted {1}- and {1, 2}-Inverses......Page 274
3. Computation of Unrestricted {1, 3}-Inverses......Page 276
4. Computation of {2}-Inverses with Prescribed Range and Null Space......Page 277
5. Greville’s Method and Related Results......Page 279
6. Computation of Least-Squares Solutions......Page 285
7. Iterative Methods for Computing A†......Page 286
Suggested Further Reading......Page 297
2. Parallel Sums......Page 298
3. The Linear Statistical Model......Page 300
4. Ridge Regression......Page 308
5. An Application of {2}-Inverses in Iterative Methods for Solving Nonlinear Equations......Page 311
6. Linear Systems Theory......Page 318
7. Application of the Group Inverse in Finite Markov Chains......Page 319
8. An Application of the Drazin Inverse to Difference Equations......Page 326
9. Matrix Volume and the Change-of-Variables Formula in Integration......Page 329
10. An Application of the Matrix Volume in Probability......Page 339
Suggested Further Reading......Page 344
2. Hilbert Spaces and Operators: Preliminaries and Notation......Page 346
3. Generalized Inverses of Linear Operators Between Hilbert Spaces......Page 352
4. Generalized Inverses of Linear Integral Operators......Page 360
5. Generalized Inverses of Linear Differential Operators......Page 364
6. Minimal Properties of Generalized Inverses......Page 372
7. Series and Integral Representations and Iterative Computation of Generalized Inverses......Page 379
8. Frames......Page 383
Suggested Further Reading......Page 384
1. Introduction......Page 386
2. The 1920 Lecture to the American Mathematical Society......Page 387
3. The General Reciprocal in General Analysis......Page 388
Bibliography......Page 391
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M......Page 428
R......Page 429
W......Page 430
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H......Page 432
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R......Page 434
V......Page 435
Z......Page 436