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广义逆的符号模式(英文版)

作者：卜长江//孙丽珠//魏益民

出版社：科学出版社出版时间：2021-06-01

开本： 16开 页数： 223

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广义逆的符号模式(英文版) 版权信息

ISBN：9787030685681
条形码：9787030685681 ; 978-7-03-068568-1
装帧：一般胶版纸
册数：暂无
重量：暂无
所属分类：
自然科学
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数学

广义逆的符号模式(英文版) 内容简介

本书讨论了广义逆符号模式的近期新发展。这些领域的根本重要性是显而易见的，因为它们与线性系统的定性分析和组合矩阵理论有关。本书提供了有关Moore-Penrose逆、Drazin逆和张量符号模式领域的介绍材料和讨论。它旨在向高年级学生和读者传达纯线性代数和应用线性代数以及组合矩阵理论的结果。

广义逆的符号模式(英文版) 目录

Contents
Preface III
Notations V
CHAPTER 1
Generalized Inverses 1
1.1 Matrix Decompositions 1
1.2 Moore-Penrose Inverse 2
1.3 Drazin Inverse 5
1.4 Group Inverse 8
1.5 Generalized Inverses and System of Linear Equations 12
1.6 Graph and Matrix 14
CHAPTER 2
Generalized Inverses of Partitioned Matrices 19
2.1 Drazin Inverse of Partitioned Matrices 19
2.2 Group Inverse of Partitioned Matrices 45
2.3 Additive Formulas for Drazin Inverse and Group Inverse 71
2.4 Drazin Inverse Index for Partitioned Matrices 96
CHAPTER 3
SNS and S2NS Matrices 101
3.1 Sign-Solvability of Linear Equations 101
3.2 Characterizations for SNS and S2NS Matrices via Digraphs 108
3.3 Ray Nonsingular and Ray S2NS Matrices 116
CHAPTER 4
Sign Pattern for Moore-Penrose Inverse 123
4.1 Least Squares Sign-Solvability 123
4.2 Matrices with Signed Moore-Penrose Inverse 125
4.3 Triangular Partitioned Matrices with Signed Moore-Penrose Inverse 138
4.4 Ray Pattern for Moore-Penrose Inverse 143
CHAPTER 5
Sign Pattern for Drazin Inverse 149
5.1 Matrices with Signed Drazin Inverse 149
5.2 Upper Triangular Partitioned Matrices with Signed Drazin Inverse 151
5.3 Anti-Triangular Partitioned Matrices with Signed Drazin Inverse 161
5.4 Bipartite Matrices with Signed Drazin Inverse 171
5.5 Sign Pattern of Group Inverse 176
5.6 Ray Pattern of Drazin Inverse 187
CHAPTER 6
Sign Pattern for Tensors 197
6.1 Tensors 197
6.2 Inverse of Tensors 200
6.3 Minimum and Maximum Rank of Sign Pattern Tensors 204
6.4 Sign Nonsingular Tensors 208
References 215
Book list of the Series in Information and Computational Science 225

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广义逆的符号模式(英文版) 节选

Chapter 1 Generalized Inverses Generalized inverses have wide applications in many fields such as numerical analysis， cryptography， operations research， probability statistics， combinatoric， optimization， astronomy， earth sciences， managerial economics， and various engineering sciences [50，56， 64， 205]. Because of the different research problems， there are many types of generalized inverses. For example， the Moore-Penrose inverse， {l}-inverse， {2}-inverse， Drazin inverse， group inverse and Bott-Duffin inverse etc. There are several monographs on generalized inverses [5，8，34，60，80，96，104，120，158，164， 193，194，196， 200， 202，207]. Other related research on perturbation analysis and preservation of generalized inverses can be found in [2， 98，117，186，213，214，224]. This chapter contains some basic concepts and tools necessary to follow the will cover some standard matrix theory tools and various graph nations and invariants. 1.1 Matrix Decompositions In this section， we list some common matrix decompositions in matrix theory. These decompositions can help readers better understand the properties of generalized inverses of matrices. Let Kmxn， Cmxn and Rmxn denote the set of all m x n matrices over skew fields K， complex field C and real field M， respectively. Let IKn， Cn and W1 denote the vector set of ?vdimensional over K， C and R，respectively. Theorem 1.1 [232]. Let A G Kmxn with rank(A) = r. Then there exist invertible matrices P and Q such that where Ir denotes an identity matrix of r-order. The decomposition of theorem 1.1 is called the equivalent decomposition of A. Theorem 1.2 [39， 232]. Let A G K . Then there exists an invertible matrix P such that where A is an invertible matrix; and iV is a nilpotent matrix. The decomposition of theorem 1.2 is called core-nilpotent decomposition of A. Next， we introduce the full rank factorization. Theorem 1.3 [8]. Let A G Kmxn with rank(i4) = r. Then there exist a full column rank B G Kmxr and a full row rank C € Krxn such that A = BC. Proof. Theorem 1.1 gives there exist invertible matrices P and Q such that Take B = P and C = [Ir 0]Q. Then A = BC is a full rank factorization For A G Cmxn， AT and A* denote the transpose and conjugate transpose of A， respectively. We know that AA* and A*A are both positive and semidefinite， and they have the same nonzero eigenvalues of AA*， where r = rank(A). Then are the singular values of A. Next is the singular value decomposition of A. Theorem 1.4 [94，212，132]. Let A G Cmxn. Then there exist unitary matrices U G Cmxm and V G Cnxn such that where is a diagonal matrix whose diagonal elements are the singular values of A. 1.2 Moore-Penrose Inverse The concept of generalized inverses was first introduced by I. Fredholm [92] in 1903 which is a particular generalized inverse of an integral operator， called pseudoin-verse. W.A. Hurwitz gave a simple algebraic construction of the pseudoinverse using the finite dimensionality of the null spaces of the Fredholm operators in 1912 [111]. The generalized inverses of differential operators appeared in D. Hilbert’s discussion of the generalized Green’s functions in 1904 [107]. In the Bulletin of the American Mathematical Society， the generalized inverses of a matrix was first introduced in 1920 by E.H. Moore [156] who is a member of the US National Academy of Sciences， where a generalized inverse is defined using projectors of matrices. In 1955， R. Penrose OM FRS， who is a foreign member of the US National Academy of Sciences and 2020 Nobel Prize in Physics， gave its equivalent definition using matrix equations [159]. The following definition of generalized inverse is given by R. Penrose. Definition 1.1 [5]. Let A G Cmxn. If a matrix X G Cnxm satisfies the following four equations AXA = A， XAX = X，(AX)* = AX， (XA)* = XA， then X is called the Moore-Penrose inverse of A (abbreviated as the M-P inverse)， denoted by A +. Obviously， if is nonsingular， then. The following theorem will elaborate the M-P inverse uniquely exists. Theorem 1.5 [5]. For a matrix A G Cmxn， A+ exists and is unique. Proof. By the singular value decomposition of matrices， there exist unitary matrices and such that ，where is a nonsingular positive diagonal matrix. Next， it can be verified that X is the M-P inverse of A. Therefore， A+ always exists. The uniqueness of A + is proved as follows. If X1 and X2 are both the M-P inverse of A， then Therefore， A+ exists and is unique. The following theorem is observed from the proof of theorem 1.5.

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