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类型 基础研究 预答辩日期 2017-12-28
开始(开题)日期 2016-06-07 论文结束日期 2017-11-08
地点 四牌楼五四楼304 论文选题来源 国家自然科学基金项目     论文字数 6 (万字)
题目 环上广义逆及相关偏序
主题词 核逆,Moore-Penrose 逆,(b,c)-逆,EP 元,偏序
摘要 矩阵的广义逆理论是矩阵理论的重要组成部分. 1955 年 R. Penrose 证明了 E. H. Moore 在 1920 年提出的矩阵广义逆概念可以用四个矩阵方程来刻画, 现称为 Moore-Penrose 逆. 1958 年 M. P. Drazin 在半群和环上引入伪逆的定义, 现称为 Drazin 逆. Moore-Penrose 逆和 Drazin 逆是两种经典的广义逆, 在许多学科有重要的应用. 随着广义逆理论的不断发展, 又出现了几类新型广义逆, 如核逆, (b,c)-逆. 环上核逆, Moore-Penrose 逆, (b,c)-逆和 EP 元是当前环上广义逆理论的热点研究问题. 由于统计研究工作的需要, 基于广义逆的偏序问题也是广义逆理论的热点研究课题. 基于广义逆的偏序主要有星偏序, 减偏序, #-偏序和核偏序. 近年来, 越来越多的学者从环的角度研究广义逆和基于广义逆的偏序问题. 由于在环上研究广义逆问题有一定的难度, 有很多问题有待解决. 因此, 本文从环的角度展开研究, 主要内容如下: 第二章考虑了环中元素的 Moore-Penrose 逆. 利用可逆元, 幂等元, 投影元和直和分解研究了环中元素的 Moore-Penrose 逆的存在性和表达式. 对于对合环 R 中的元素 a, 得到了环中元素有 Moore-Penrose 逆的如下判断准则: a 是 Moore-Penrose 可逆的当且仅当 a 是良好支撑元, 从而说明 J. J. Koliha 等人发表在 Linear Algebra Appl. 上关于 Moore-Penrose 逆等价刻画里 *-可消条件可以去掉. 其次, 利用直和分解讨论了 Moore-Penrose 逆存在的充要条件. 最后, 研究了基于 Moore-Penrose 逆的投影元的相关性质. 第三章主要研究了环中元素的核逆. 首先讨论了核逆的判别准则, 利用三个方程给出了核逆的存在的充要条件, 从而简化了D. S. Rakic 等人发表在 Linear Algebra Appl. 上关于核逆的五个方程的刻画. 回答了一个群可逆元何时是核可逆的问题. 利用可逆元刻画了核逆的存在性. 其次讨论了核逆的吸收律和反序律成立的条件, 以及环上三角矩阵的核可逆性问题. 最后研究了在一定条件下两个核可逆元和的核可逆性问题, 并给出表达式. 第四章主要研究了环中元素 (b,c)-逆的存在性和表达式, 应用中心化子讨论了 (b,c)-逆的吸收律和反序律问题. 由于 (b,c)-逆是 {2}-逆, 但一般不是 {1}-逆, 本章讨论了 (b,c)-逆何时是 {1}-逆的问题, 证明了对于 R 中元素a,b,c,y, y 是 a 的内 (b,c)-逆当且仅当 a 是正则的, R=a^{\circ}\oplus bR 和 R={}^{\circ}a\oplus Rc. 从而推广了关于~Moore-Penrose 逆, 群逆和核逆的相关结果. 最后, 研究了交换子 A \bcf{A}-\bcf{A}A 的秩, 给出交换子 A \bcf{A}-\bcf{A}A 的如下秩等式, 其中 A,B,C\in \fu^{n\times n}. 若 A 是 (B,C)-可逆的且 \bcf{A} 是 A 的 (B,C)-逆, 则有 \rk{(A \bcf{A}-\bcf{A}A)}=\rk{\left(\lie{CA}{C}\right)}+\rk{([A B\ | \ B])}-2\rk{(B)}, 从而改进了刘永辉等人的相关结果. 第五章讨论了环中的 EP元的等价刻画. 首先给出 EP元的方程刻画. 证明了对于 a\in R, a 是 EP 元当且仅当存在 x\in R 使得 (xa)^{\ast}=xa, xa^{2}=a 和 ax^{2}=x. 其次回答了核可逆元 (Moore-Penrose 可逆元, 群可逆元) 何时为 EP元的问题. 对于 a\in R, 将 P. Patricio 和 R. Puystjens 发表在 Linear Algebra Appl. 上关于 EP 元等价刻画的条件 aR=a^{\ast}R 弱化为 aR\subseteq a^{\ast}R. 最后, 引进了n-EP 元的定义, 推广了双-EP 的概念, 给出了~EP 元的等价刻画. 第六章主要研究了基于广义逆的星偏序, 核偏序, 钻石偏序等偏序. 对两个 Moore-Penrose 可逆元, 在一定条件下讨论了星偏序的伪上半格存在性, 从而改进了 R. E. Hartwig 发表在 Proc. Amer. Math. Soc. 上的结果. 其次, 讨论了核偏序和左星偏序, 钻石偏序和左星偏序等基于广义逆的偏序之间的联系. 最后, 更正了 S. B. Malik 发表在Appl. Math. Comput. 上的文章的错误.
英文题目 Generalized inverses and related partial orders over rings
英文主题词 Core inverse, Moore-Penrose inverse, (b,c)-inverse, EP element, partial order
英文摘要 The generalized inverse theory of matrices is an important subject of the study of matrices. The first concept of generalized inverse was introduced by E. H. Moore in 1920 and a simpler characterization of this generalized inverse was given in terms of four matrix equations by R. Penrose in 1955. Three years later, Drazin introduced another kind of generalized inverse in semigroups and rings, which is called Drazin inverse nowdays. The Moore-Penrose inverse and the Drazin inverse are two classic generalized inverses. With the development of science and technology, the theory of generalized inverses of matrices has got deep development and extensive applications. In recent years, some new concepts of generalized inverses were introduced, for example, the core inverse and (b,c)-inverse. The core inverse, Moore-Penrose inverse, (b,c)-inverse and EP element are hot topics in generalized inverses research field. Due to the applications in statistics, partial order theory based on generalized inverses also is a hot topic in generalized inverses research field. For example, the star partial order, the minus partial order, the sharp partial order and the core partial order have been introduced. In recent years, more and more scholars study the generalized inverses and partial orders based on generalized inverses over a ring. In this thesis, we mainly adopt some ring theoretic methods to study the theory of generalized inverses in rings. The main contents are arranged as follows: In chapter 2, existence criteria and expressions for the Moore-Penrose inverse are characterized by invertible, Hermite, projection elements and direct sums. we show that an element a in a ring R with involution is Moore-Penrose invertible if and only if a is well-supported. Therefore, the star-cancelable condition of the work of J. J. Koliha et al. can be dropped. Also, we show that the existence of the Moore-Penrose inverse can be characterized by some decompositions of the ring. In addition, the formulae of the Moore-Penrose inverse of an element in R are presented. Furthermore, some properties of projections related the Moore-Penrose inverse are investigated. In chapter 3, we investigate the core inverse over a ring with an involution. Firstly, we study the existence of the core inverses in terms of equations. We prove that the existence criterion of the core inverse can be characterized by three equations, which improved the work of D. S. Rakic et al. in Linear Algebra Appl. We answer when a group invertible element is core invertible. We also use invertible elements to characterize the existence of the core inverses. Secondly, several necessary and sufficient conditions which guarantee the absorption law and the reverse order law for core invertible elements hold are given. We also consider the core invertibility of triangular matrices over a ring with an involution. Finally, we investigate the additive property of two core invertible elements. Moreover, the formulae of the sum of two core invertible elements are presented. In chapter 4, existence criteria for the (b,c)-inverse are obtained and centralizer’s applications of the (b,c)-inverse are considered. We present explicit expressions for the (b,c)-inverse in terms of inner inverses. Let a,b,c\in R, it is well-known that the (b,c)-inverse of a is an outer {2}-inverse of a, but not an inner inverse of a. We answer the question when the (b,c)-inverse of a\in R is an inner inverse of a. Let a,b,c,y\in R. We show that y is an inner (b,c)-inverse of a if and only if a is regular, R=a^{\circ}\oplus bR and R={}^{\circ}a\oplus Rc. Moreover, rank equalities of the commutator A\bcf{A}-\bcf{A}A are obtained, where A,B,C\in \fu^{n\times n}. If A is (B,C)-invertible with (B,C)-inverse \bcf{A}, then we have \rk{(A \bcf{A}-\bcf{A}A)}=\rk{\left(\lie{CA}{C}\right)}+\rk{([A B\ | \ B])}-2\rk{(B)}, which is an improvement of the works of Yonghui Liu et al. In chapter 5, we characterize the EP elements in a ring R by three equations. Namely, if a\in R, then a is EP if and only if there exists x\in R such that (xa)^{\ast}=xa, xa^{2}=a and ax^{2}=x. Many equivalent conditions for a core (Moore-Penrose) invertible element to be an EP element are given. We show that the condition aR=a^\ast R in the work of P. Patricio and R. Puystjens on Linear Algebra Appl. can be relaxed as aR\subseteq a^\ast R. Finally, any EP element is characterized in terms of the n-EP property, which is a generalization of the bi-EP property. In chapter 6, several characterizations and properties of core partial order, star partial order and diamond partial order in $R$ are given. We show that the pseudo upper semilattice of two Moore-Penrose invertible elements exists under some conditions. Also, the expression of such pseudo upper semilattice is obtained. This is an improvement of the work of R. E. Hartwig in Proc. Amer. Math. Soc. Moreover, the relationships among the core partial order and left star partial order, the diamond partial order and left star partial order are obtained. Finally, we corrected two errors of the work of S. B. Malik in Appl. Math. Comput.
学术讨论
主办单位时间地点报告人报告主题
东南大学 2014.09.19 数学学院 第一报告厅 D.S.Cvetkovic-Ilic Generalized inverses, projectors, completions of operator matrices
东南大学 2015.11.26 数学学院 第二报告厅 魏益民 Perturbation Bounds of Tensor Eigen values and Singular Values Problems with Even Order
东南大学 2016.04.20 数学学院 第一报告厅 许三长 Projections for generalized inverses
东南大学 2016.06.22 数学学院 第一报告厅 许三长 New characterizations for core inverses in rings with involution
瓦伦西亚理工大学 2016.10.27 数学楼318室 许三长 Partial order base on the CS decomposition
瓦伦西亚理工大学 2017.06.21 数学楼318室 许三长 On characterizations of special elements in rings with involution
东南大学 2017.10.31 数学系第一报告厅 黄强联 The Stable Perturbation of Generalized Inverses
东南大学 2017.11.01 数学学院第一报告厅 许庆祥 An introduction to Hilbert C*-module
     
学术会议
会议名称时间地点本人报告本人报告题目
第七届中日韩国际环论会议 2015年07月01日至07日 浙江工业大学 The Moore-Penrose inverse in rings
第十四届全国代数学学术会议 2016年05月26日至31日 扬州大学 EP elements in rings with involution
广义逆理论研讨会 2016年12月21日至22日 瓦伦西亚理工大学 Generalized core inverses of matrices
(b,c)-逆理论和应用研讨会 2016年06月12日至15日 瓦伦西亚理工大学 Existence criteria and expressions of the (b, c)-inverse in rings and its applications
     
代表作
论文名称
New characterizations for core inverses in rings with involution
Existence Criteria and Expressions of the (b, c)-Inverse in Rings and Their Applications
 
答辩委员会组成信息
姓名职称导师类别工作单位是否主席备注
秦厚荣 正高 教授 博导 南京大学
丁南庆 正高 教授 博导 南京大学
朱晓胜 正高 教授 博导 南京大学
黄兆泳 正高 教授 博导 南京大学
张良云 正高 教授 博导 南京农业大学
王栓宏 正高 教授 博导 东南大学
      
答辩秘书信息
姓名职称工作单位备注
王周 其他 副教授 东南大学