Contents
1 Equation Solving Generalized Inverses 1
1.1 Moore-Penrose Inverse 1
1.1.1 Definition and Basic Properties of At 2
1.1.2 Range and Null Space of a Matrix4
1.1.3 Full-Rank Factorization.5
1.1.4 Minimum-Norm Least-Squares Solution 7
1.2 The{i j, k} Inverses10
1.2.1 The {1} Inverse and the Solution of a Consistent System of Linear Equations 10
1.2.2 The {1,4} Inverse and the Minimum-Norm Solution of a Consistent System 11
1.2.3 The {1,3} Inverse and the Least-Squares Solution of An Inconsistent System 12
1.2.4 The {1} Inverse and the Solution of the Matrix Equation A×B = D 14
1.2.5 The {1} Inverse and the Common Solution ofAx-aandBx-b15
1.2.6 The {1} Inverse and the Common Solution ofAX -BandXD-E 18
1.3 The Generalized Inverses With Prescribed Range and Null Space 19
1.3.1 Idempotent Matrices and Projectors 20
1.3.2 Generalized Inverse A7-tls2).25
1.3.3 Urquhart Formula 28
1.3.4 Generalized Inverse Art21_ 31
1.4 Weighted Moore-Penrose Inverse 33
1.4.1 Weighted Norm and Weighted Conjugate Transpose Matrix.34
1.4.2 The {1,4N} Inverse and the Minimum-Norm (N) Solution of a Consistent System of Linear Equations 37
1.4.3 The {1,3M} Inverse and the Least-Squares (M) Solution of An Inconsistent System of Linea \iear Equations 38
1.4.4 Weighted Moore-Penrose Inverse and The Minimum-Norm (N) and Least-Squares (M) Solution of An Inconsistent System of Linear Equations 39
1.5 Bott-Duffin Inverse and Its Generalization42
1.5.1 Bott-Duffin Inverse and the Solution of Constrained Linear Equations42
1.5.2 The Necessary and Sufficient Conditions for the Existence of the Bott-Duffin Inverse 45
1.5.3 Generalized Bott-Duffin Inverse and Its Properties 49
1.5.4 The Generalized Bott-Duffin Inverse and the Solution of Linear Equations 58
References 63
2 Drazin Inverse 65
2.1 Drazin Inverse 66
2.1.1 Matrix Index and Its Basic Properties 66
2.1.2 Drazin Inverse and Its Properties 67
2.1.3 Core-Nilpotent Decomposition 73
2.2 Group Inverse 75
2.2.1 Definition and Properties of the Group Inverse 76
2.2.2 Spectral Properties of the Drazin and Group
2.3 W-Weighted Drazin Inverse83
References.89
3 Generalization of the Cramer's Rule and the Minors of the Generalized Inverses 91
3.1 Nonsingularity of Bordered Matrices 92
3.1.1 Relations with ALN and At 92
3.1.2 Relations Between the Nonsingularity of Bordered Matrices and Ad and Ag 94
3.1.3 Relations Between the Nonsingularity of Bordered Matrices and A7t2~, A-r~ls2)and A}云” 96
3.2 Cramer's Rule for Solutions of Linear Systems 98
3.2.1 Cramer's Rule for the Minimum-Norm (N) Least-Squares (M) Solution of an Inconsistent System of Linear Equations 99
3.2.2 Cramer's Rule for the Solution of a Class of Singular Linear Equations 101
3.2.3 Cramer's Rule for the Solution of a Class of Restricted Linear Equations.103
3.2.4 An Alternative and Condensed Cramer's Rule for the Restricted Linear Equations.107
3.3 Cramer's Rule for Solution of a Matrix Equation.114
3.3.1 Cramer's Rule for the Solution of a Nonsingular Matrix Equation114
3.3.2 Cramer's Rule for the Best-Approximate Solution of a Matrix Equation.116
3.3.3 Cramer's Rule for the Unique Solution of a Restricted Matrix Equation120
3.3.4 An Alternative Condensed Cramer's Rule for a Restricted Matrix Equation126
3.4 Determinantal Expressions of the Generalized Inverses and Projectors 128
3.5 The Determinantal Expressions of the Minors of the Generalized Inverses.131
3.5.1 Minors of the Moore-Penrose Inverse134
3.5.2 Minors of the Weighted Moore-Penrose Inverse.137
3.5.3 Minors of the Group Inverse and Drazin Inverse142
3.5.4 Minors of the Generalized Inverse ArL2s) 147
References 149
4 Reverse Order and Forward Order Laws for Ar~2s)153
4.1Introduction 153
4.2 Reverse Order Law 159
4.3 Forward Order Law 164
References 173
5 Computational Aspects 175
5.1 Methods Based on the Full Rank Factorization 176
5.1.1 Row Echelon Forms178
5.1.2 Gaussian Elimination with Complete Pivoting180
5.1.3 Householder Transformation 182
5.2 Singular Value Decompositions and (M,N) Singular Value Decompositions.185
5.2.1 Singular Value Decomposition.185
5.2.2 (M,N) Singular Value Decomposition 187
5.2.3 Methods Based on SVD and (M,N) SVD 189
5.3 Generalized Inverses of Sums and Partitioned Matrices193
5.3.1 Moore-Penrose Inverse of Rank-One Modified Matrix 194
5.3.2 Greville's Method 200
5.3.3 Cline's Method 203
5.3.4 Noble's Method 205
5.4 Embedding Methods 211
5.4.1 Generalized Inverse as a Limit 211
5.4.2 Embedding Methods 214
5.5 Finite Algorithms 217
References 222
6 Structured Matrices and Their Generalized Inverses 225
6.1 Computing the Moore-Penrose Inverse of a Toeplitz
6.2 Displacement Structure of the Generalized Inverses 228
References 231
7 Parallel Algorithms for Computing the Generalized Inverses 233
7.1 The Model of Parallel Processors 234
7.1.1 Array Processor 234
7.1.2 Pipeline Processor 234
7.1.3 Multiprocessor 235
7.2 Measures of the Performance of Parallel Algorithms 237
7.3 Parallel Algorithms 238
7.3.1 Basic Algorithms 239
7.3.2 Csanky Algorithms 248
7.4 Equivalence Theorem 255
References 260
8 Perturbation Analysis of the Moore-Penrose Inverse and the Weighted Moore-Penrose Inverse 263
8.1 Perturbation Bounds 263
8.2 Continuity 274
8.3 Rank-Preserving Modification 276
8.4 Condition Numbers 278
8.5 Expression for the Perturbation of Weighted Moore-Penrose Inverse282
References 288
9 Perturbation Analysis of the Drazin Inverse and the Group Inverse 291
9.1 Perturbation Bound for the Drazin Inverse 291
9.2 Continuity of the Drazin Inverse 294
9.3 Core-Rank Preserving Modification of Drazin Inverse 296
9.4 Condition Number of the Drazin Inverse 298
9.5 Perturbation Bound for the Group Inverse 300
References 305
10 Generalized Inverses of Polynomial Matrices 307
10.1 Introduction 307
10.2 Moore-Penrose Inverse of a Polynomial Matrix 309
10.3 Drazin Inverse of a Polynomial Matrix 311
References 316
11 Moore-Penrose Inverse of Linear Operators 317
11.1 Definition and Basic Properties 317
11.2 Representation Theorem325
11.3 Computational Methods 327
11.3.1 Euler-Knopp Methods 327
11.3.2 Newton Methods 328
11.3.3 Hyperpower Methods 330
11.3.4 Methods Based on Interpolating Function References 337
12 0perator Drazin Inverse 339
12.1 Definition and Basic Properties 339
12.2 Representation Theorem 343
12.3 Computational Procedures 346
12.3.1 Euler-Knopp Method 346
12.3.2 Newton Method 347
12.3.3 Limit Expression 348
12.3.4 Newton Interpolation 349
12.3.5 Hermite Interpolation 350
12.4 Perturbation Bound 352
12.5 Weighted Drazin Inverse of an Operator 355
12.5.1 Computational Methods 360
12.5.2 Perturbation Analysis 367
References 371