1 Equation Solving Generalized Inverses 1.1 Moore-Penmse Inverse 1.1.1 Definition and Basic Properties of At 1.1.2 Range and Null Space of a Matrix 1.1.3 Full-Rank Factorization 1.1.4 Minimum-Norm Least-Squares Solution 1.2 The {i,j, k} Inverses 1.2.1 The {1} Inverse and the Solution of a Consistent System of Linear Equations 1.2.2 The {1,4} Inverse and the Minimum-Norm Solution of a Consistent System 1.2.3 The {1, 3} Inverse and the Least-Squares Solution of An Inconsistent System 1.2.4 The {1} Inverse and the Solution of the Matrix Equation AX B = D 1.2.5 The {1} Inverse and the Common Solution of Ax = a and Bx = b 1.2.6 The {1} Inverse and the Common Solution of AX = B and XD = E 1.3 The Generalized Inverses With Prescribed Range and Null Space 1.3.1 Idempotent Matrices and Projectors a(1,2) 1.3.2 Generalized Inverse A(1.2)T,S 1.3.3 Urquhart Formula 1.3.4 Generalized Inverse a(2)T,S 1.4 Weighted Moore-Penrose Inverse 1.4.1 Weighted Norm and Weighted Conjugate Transpose Matrix 1.4.2 The {1,4N} Inverse and the Minimum-Norm (N) Solution of a Consistent System of Linear Equations 1.4.3 The {1, 3M} Inverse and the Least-Squares (M) Solution of An Inconsistent System of Linear Equations 1.4.4 Weighted Moore-Penrose Inverse and The Minimum-Norm (N) and Least-Squares (M) Solution of An Inconsistent System of Linear Equations 1.5 Bott-Duffin Inverse and Its Generalization 1.5.1 Bott-Duffin Inverse and the Solution of Constrained Linear Equations 1.5.2 The Necessary and Sufficient Conditions for the Existence of the Bott-Duffin Inverse 1.5.3 Generalized Bott-Duffin Inverse and Its Properties 1.5.4 The Generalized Bott-Duffin Inverse and the Solution of Linear Equations References 2 Drazin Inverse 2.1 Drazin Inverse 2.1.1 Matrix Index and Its Basic Properties 2.1.2 Drazin Inverse and Its Properties 2.1.3 Core-Nilpotent Decomposition 2.2 Group Inverse 2.2.1 Definition and Properties of the Group Inverse 2.2.2 Spectral Properties of the Drazin and Group Inverses 2.3 W-Weighted Drazin Inverse References 3 Generalization of the Cramer's Rule and the Minors of the Generalized Inverses 3.1 Nonsingularity of Bordered Matrices 3.1.1 Relations with A MN and A 3.1.2 Relations Between the Nonsingularity of Bordered Matrices and Ad and Ag 3.1.3 Relations Between the Nonsingularity of Bordered Matrices and A(2)T,S��A(l'2)T,S�� and A(-1)(L) 3.2 Cramer's Rule for Solutions of Linear Systems 3.2.1 Cramer's Rule for the Minimum-Norm (N) Least-Squares (M) Solution of an Inconsistent System of Linear Equations 3.2.2 Cramer's Rule for the Solution of a Class of Singular Linear Equations 3.2.3 Cramer's Rule for the Solution of a Class of Restricted Linear Equations 3.2.4 An Alternative and Condensed Cramer's Rule for the Restricted Linear Equations 3.3 Cramer's Rule for Solution of a Matrix Equation 3.3.1 Cramer's Rule for the Solution of a Nonsingular Matrix Equation 3.3.2 Cramer's Rule for the Best-Approximate Solution of a Matrix Equation 3.3.3 Cramer's Rule for the Unique Solution of a Restricted Matrix Equation 3.3.4 An Alternative Condensed Cramer's Rule for a Restricted Matrix Equation 3.4 Determinantal Expressions of the Generalized Inverses and Projectors 3.5 The Determinantal Expressions of the Minors of the Generalized Inverses 3.5.1 Minors of the Moore-Penrose Inverse 3.5.2 Minors of the Weighted Moore-Penrose Inverse 3.5.3 Minors of the Group Inverse and Drazin Inverse 3.5.4 Minors of the Generalized Inverse A(2)T,S References 4 Reverse Order and Forward Order Laws for A(2)T,S 4.1 Introduction 4.2 Reverse Order Law 4.3 Forward Order Law References 5 Computational Aspects 5.1 Methods Based on the Full Rank Factorization 5.1.1 Row Echelon Forms 5.1.2 Gaussian Elimination with Complete Pivoting 5.1.3 Householder Transformation 5.2 Singular Value Decompositions and (M, N) Singular Value Decompositions 5.2.1 Singular Value Decomposition 5.2.2 (M, N) Singular Value Decomposition 5.2.3 Methods Based on SVD and (M, N) SVD 5.3 Generalized Inverses of Sums and Partitioned Matrices 5.3.1 Moore-Penrose Inverse of Rank-One Modified Matrix 5.3.2 Greville's Method 5.3.3 Cline's Method 5.3.4 Noble's Method 5.4 Embedding Methods 5.4.1 Generalized Inverse as a Limit 5.4.2 Embedding Methods 5.5 Finite Algorithms References 6 Structured Matrices and Their Generalized Inverses 6.1 Computing the Moore-Penrose Inverse of a Toeplitz Matrix 6.2 Displacement Structure of the Generalized Inverses References 7 Parallel Algorithms for Computing the Generalized Inverses 7.1 The Model of Parallel Processors 7.1.1 Array Processor 7.1.2 Pipeline Processor 7.1.3 Multiprocessor 7.2 Measures of the Performance of Parallel Algorithms 7.3 Parallel Algorithms 7.3.1 Basic Algorithms 7.3.2 Csanky Algorithms 7.4 Equivalence Theorem References 8 Perturbation Analysis of the Moore-Penrose Inverse and the Weighted Moore-Penrose Inverse 8.1 Perturbation Bounds 8.2 Continuity 8.3 Rank-Preserving Modification 8.4 Condition Numbers 8.5 Expression for the Perturbation of Weighted Moore-Penrose Inverse References 9 Perturbation Analysis of the Drazin Inverse and the Group Inverse 9.1 Perturbation Bound for the Drazin Inverse 9.2 Continuity of the Drazin Inverse 9.3 Core-Rank Preserving Modification of Drazin Inverse 9.4 Condition Number of the Drazin Inverse 9.5 Perturbation Bound for the Group Inverse References 10 Generalized Inverses of Polynomial Matrices 10.1 Introduction 10.2 Moore-Penrose Inverse of a Polynomial Matrix 10.3 Drazin Inverse of a Polynomial Matrix References 11 Moore-Penrose Inverse of Linear Operators 11.1 Definition and Basic Properties 11.2 Representation Theorem 11.3 Computational Methods 11.3.1 Euler-Knopp Methods 11.3.2 Newton Methods 11.3.3 Hyperpower Methods 11.3.4 Methods Based on Interpolating Function Theory References 12 Operator Drazin Inverse 12.1 Definition and Basic Properties 12.2 Representation Theorem 12.3 Computational Procedures 12.3.1 Euler-Knopp Method 12.3.2 Newton Method 12.3.3 Limit Expression 12.3.4 Newton Interpolation 12.3.5 Hermite Interpolation 12.4 Perturbation Bound 12.5 Weighted Drazin Inverse of an Operator 12.5.1 Computational Methods 12.5.2 Perturbation Analysis References Index