廣義逆：理論與計算（英文版，第二版）

《廣義逆：理論與計算（英文版，第二版）》是2018年科學出版社出版的圖書，作者是王國榮、魏益民、喬三正。

基本介紹

• 書名：廣義逆：理論與計算（英文版，第二版）
• 作者：王國榮、魏益民、喬三正
• 出版社：科學出版社
• 出版時間：2018年01月01日
• ISBN：9787030595645

內容簡介

《廣義逆：理論與計算（第2版 英文版）》從廣義逆的基本原理開始，然後轉向更高級的主題。對克拉默定律的推廣、廣義逆的行列式表示、矩陣乘積廣義逆的反序律、結構矩陣廣義逆的結構、廣義逆的並行計算、廣義逆的攝動分析進行了理論研究。逆，廣義逆、廣義奇異值分解、嵌入法、有限法、多項式矩陣的廣義逆和線性運算元的廣義逆的全秩分解計算方法的算法研究。

《廣義逆：理論與計算（第2版 英文版）》是針對研究人員，博士後，和研究生領域的廣義逆與大學水平的理解線性代數。

圖書目錄

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

……

10Generalized Inverses of Polynomial Matrices

11Moore-Penrose Inverse of Linear Operators

12Operator Drazin Inverse

Index