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

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