广义二次矩阵的若干研究

发布时间：2018-03-24 04:12

本文选题：广义二次矩阵　切入点：数量三幂等矩阵　出处：《福建师范大学》2016年博士论文

【摘要】：上个世纪以来,具有幂条件的矩阵类及其线性组合的性质研究一直是矩阵代数的重要课题之一.随着研究的不断深入,2005年R.W.Farebrother及G.Trenkler引入了具有广泛意义的广义二次矩阵(见[39]),概括统一了已有丰富研究成果的对合矩阵、幂等矩阵及算子等.这一基础性的研究引起了不少人对广义二次矩阵的关注.广义二次矩阵在概率统计、密码学、控制理论、量子力学和很多数学、物理领域有着重要的应用.本文利用广义二次矩阵的等价定义,进一步研究广义二次矩阵的基本性质,得到矩阵方程aA+bX=AX的广义二次解,确定数量三幂等矩阵与广义二次矩阵间的关系，并给出广义二次矩阵在若干运算下的秩等式及应用等内容.这些结果将丰,富和深化二次矩阵及二次算子的理论研究，为进一步的讨论提供强有力的工具.全文具体结构如下：绪论部分对与本文有关的幂等矩阵、对合矩阵等常见的具有幂条件的矩阵类的研究进行综述.回顾广义二次矩阵的发展历史以及研究现状,叙述了本文的研究内容及论文框架.在R.W.Farebrother及G.Trenkler研究的基础上.第一章进一步讨论广义二次矩阵的基本性质.从广义二次矩阵的表示、相似标准形、方幂、秩、逆及广义逆等方面进行讨论.得到一些更深刻的结果，如：给出广义二次矩阵的方幂A~k的显式表达,指出[39]中关于可逆广义二次矩阵的逆的讨论中的问题,求出可逆广义二次矩阵的所有逆矩阵，证明了广义二次矩阵的方幂及逆还是广义二次的,且清晰地表示了A~k、A~(-1)与A广义二次性特征间的关系.最后求出了广义二次矩阵的所有{1}及{1,2}-广义逆及群逆.注意到相关矩阵方程也是矩阵讨论中的一个基本问题，为得到矩阵方程aA+bX=AX有广义二次解的充要条件，第二章介绍了线性组合与积相等的矩阵对的研究现状.并探讨线性组合与积相等的矩阵对在特征值、可逆性、广义二次性方面的密切联系，这也给出广义二次矩阵的和、积仍为广义二次的又一个充分条件.第三章考虑了数量三幂等矩阵是广义二次矩阵的情况,给出数量三幂等矩阵的分类,还给出了任意有限多个数量三幂等矩阵和的秩等式.由于幂等矩阵是数量三幂等的,作为应用,解决了Y.Tian和G.P.H.Stvan提出的关于任意有限多个幂等矩阵和的秩等式的公开问题.本章的最后还得到了任意有限多个广义二次矩阵和的秩等式.借助广义二次矩阵与幂等矩阵的密切关系,第四章致力于研究广义二次矩阵在线性运算及其组合下的秩等式,得到广义二次矩阵和与积的线性组合的秩与零度的不变性、广义二次矩阵换位子的秩等式及广义二次矩阵的广义Jordan积的秩的不变性,并给出许多应用，概括了J.Gr(?)β, G.Trenkler, J.J.Koliha, Y.Tian, G.P.H.Styan等关于幂等矩阵、对合矩阵的相关结果.最后给出一个秩等式,它统一了矩阵秩不变性的相关等式.最后，，对本文的研究工作进行了总结和展望，指出本文的许多结论可以在广义二次算子上进一步延伸。
[Abstract]:Since the last century, study the properties of matrices with power condition and the linear combination has been one of the important topics of matrix algebra. With the deepening of research, 2005 R.W.Farebrother and G.Trenkler introduced the generalized two matrix broad sense (see [39]), summarized existing research results enrich the unified involutory matrix, idempotent matrix and operator. On this basis caused a lot of people focus on the two generalized matrix. Two generalized matrix in probability and statistics, cryptography, control theory, quantum mechanics and mathematical physics, field has important applications. This paper uses the equivalent definition of generalized two matrix, the basic properties of further research the two generalized matrix, obtained the generalized matrix equation aA+bX=AX two solution, determine the relationship between the number of three idempotent matrices and the generalized two times between the matrix, and gives the generalized two matrix in the if Dry rank equality and application content under the operation. These results will be abundant, rich and deepen the theoretical research of two times and two times the matrix operator, provide a powerful tool for further discussion. The concrete structure is as follows: the introduction of idempotent matrix associated with this article, we summarize the study matrices of involutory matrix etc. familiar with power condition. The development history and research status of the two review of generalized matrix, introduces the research content and the frame of this paper. Based on the study of R.W.Farebrother and G.Trenkler. The first chapter discussed the basic properties of generalized matrix two. Two said from the generalized matrix, similar to the standard form of power. The discussion, rank, inverse and generalized inverse and so on. Get some more profound results, such as: the explicit expression of power A~k generalized two matrix, points out that the [39] on the two generalized reversible matrix The inverse problems in the discussion, find out all the two generalized inverse matrix invertible matrix, proves that the power of the two generalized matrix and generalized inverse is two times, and clearly expressed A~k, A~ (-1) and A generalized characteristic two times between the relationship. Finally the all {1} and {1,2}- generalized two matrix generalized inverse and group inverse. Note that one of the basic problems related to the matrix equation is also discussed in the matrix, in order to obtain the matrix equation aA+bX=AX has generalized two solvability conditions, the second chapter introduces the current situation of the research and the linear combination of the same product of matrix. And to explore the linear matrix. In combination with the same product of on eigenvalue, reversible, generalized two aspects of close contact, this also gives the generalized matrix and two times, two times the product is generalized and a sufficient condition. The third chapter considers the number three is the two generalized idempotent matrix matrix, to The classification number of three idempotent matrix, rank equality and arbitrary finite number of three idempotent matrices is given. The idempotent matrix is idempotent number three, as an application, to solve the Y.Tian and G.P.H.Stvan concerning rank equalities and arbitrary finite number of idempotent matrix of open questions at the end of this chapter. The rank equalities and arbitrary finite number of generalized matrix two. With the close relationship between the two generalized matrix and idempotent matrix, the fourth chapter is devoted to the study of generalized matrix in linear operation and two combinations of the rank equalities, get two invariant linear combination of generalized matrix and product rank with zero the invariance of the generalized Jordan equation of the rank two matrix generalized commutator and generalized two matrix product rank, and gives many applications, summarizes the J.Gr (?) G.Trenkler, J.J.Koliha, beta, Y.Tian, G.P.H.Styan and so on power The related results of equal matrix and involution matrix are given. Finally, a rank equation is presented, which unifies the related equation of rank invariance of the matrix. Finally, the research work in this paper is summarized and prospected, and many conclusions can be further extended to the generalized two operator.

【学位授予单位】：福建师范大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O151.21

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