本文選題：廣義二次矩陣　切入點：數(shù)量三冪等矩陣　出處：《福建師范大學》2016年博士論文

【摘要】：上個世紀以來,具有冪條件的矩陣類及其線性組合的性質(zhì)研究一直是矩陣代數(shù)的重要課題之一.隨著研究的不斷深入,2005年R.W.Farebrother及G.Trenkler引入了具有廣泛意義的廣義二次矩陣(見[39]),概括統(tǒng)一了已有豐富研究成果的對合矩陣、冪等矩陣及算子等.這一基礎(chǔ)性的研究引起了不少人對廣義二次矩陣的關(guān)注.廣義二次矩陣在概率統(tǒng)計、密碼學、控制理論、量子力學和很多數(shù)學、物理領(lǐng)域有著重要的應用.本文利用廣義二次矩陣的等價定義,進一步研究廣義二次矩陣的基本性質(zhì),得到矩陣方程aA+bX=AX的廣義二次解,確定數(shù)量三冪等矩陣與廣義二次矩陣間的關(guān)系，并給出廣義二次矩陣在若干運算下的秩等式及應用等內(nèi)容.這些結(jié)果將豐,富和深化二次矩陣及二次算子的理論研究，為進一步的討論提供強有力的工具.全文具體結(jié)構(gòu)如下：緒論部分對與本文有關(guān)的冪等矩陣、對合矩陣等常見的具有冪條件的矩陣類的研究進行綜述.回顧廣義二次矩陣的發(fā)展歷史以及研究現(xiàn)狀,敘述了本文的研究內(nèi)容及論文框架.在R.W.Farebrother及G.Trenkler研究的基礎(chǔ)上.第一章進一步討論廣義二次矩陣的基本性質(zhì).從廣義二次矩陣的表示、相似標準形、方冪、秩、逆及廣義逆等方面進行討論.得到一些更深刻的結(jié)果，如：給出廣義二次矩陣的方冪A~k的顯式表達,指出[39]中關(guān)于可逆廣義二次矩陣的逆的討論中的問題,求出可逆廣義二次矩陣的所有逆矩陣，證明了廣義二次矩陣的方冪及逆還是廣義二次的,且清晰地表示了A~k、A~(-1)與A廣義二次性特征間的關(guān)系.最后求出了廣義二次矩陣的所有{1}及{1,2}-廣義逆及群逆.注意到相關(guān)矩陣方程也是矩陣討論中的一個基本問題，為得到矩陣方程aA+bX=AX有廣義二次解的充要條件，第二章介紹了線性組合與積相等的矩陣對的研究現(xiàn)狀.并探討線性組合與積相等的矩陣對在特征值、可逆性、廣義二次性方面的密切聯(lián)系，這也給出廣義二次矩陣的和、積仍為廣義二次的又一個充分條件.第三章考慮了數(shù)量三冪等矩陣是廣義二次矩陣的情況,給出數(shù)量三冪等矩陣的分類,還給出了任意有限多個數(shù)量三冪等矩陣和的秩等式.由于冪等矩陣是數(shù)量三冪等的,作為應用,解決了Y.Tian和G.P.H.Stvan提出的關(guān)于任意有限多個冪等矩陣和的秩等式的公開問題.本章的最后還得到了任意有限多個廣義二次矩陣和的秩等式.借助廣義二次矩陣與冪等矩陣的密切關(guān)系,第四章致力于研究廣義二次矩陣在線性運算及其組合下的秩等式,得到廣義二次矩陣和與積的線性組合的秩與零度的不變性、廣義二次矩陣換位子的秩等式及廣義二次矩陣的廣義Jordan積的秩的不變性,并給出許多應用，概括了J.Gr(?)β, G.Trenkler, J.J.Koliha, Y.Tian, G.P.H.Styan等關(guān)于冪等矩陣、對合矩陣的相關(guān)結(jié)果.最后給出一個秩等式,它統(tǒng)一了矩陣秩不變性的相關(guān)等式.最后，，對本文的研究工作進行了總結(jié)和展望，指出本文的許多結(jié)論可以在廣義二次算子上進一步延伸。
[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.

【學位授予單位】：福建師范大學
【學位級別】：博士
【學位授予年份】：2016
【分類號】：O151.21

【參考文獻】

相關(guān)期刊論文 前10條

1 周士藩;;在左主理想環(huán)上的Cochran定理[J];浙江師范學院學報(自然科學版);1984年01期

2 林文元,林春土;關(guān)于Cochran定理的推廣[J];高校應用數(shù)學學報A輯(中文版);1989年04期

3 方開泰;吳月華;;二次型分布與COCHRAN定理[J];經(jīng)濟數(shù)學;1984年00期

4 徐兆亮,王國榮;關(guān)于冪等矩陣和對合矩陣的幾個結(jié)果[J];上海海運學院學報;2003年02期

5 劉玉,曹重光;體上某些分塊矩陣的Drazin逆(英文)[J];黑龍江大學自然科學學報;2004年04期

6 段櫻桃;杜鴻科;;關(guān)于廣義二次算子(英文)[J];應用泛函分析學報;2007年01期

7 劉玉;曹重光;;關(guān)于除環(huán)上矩陣秩的幾個等式[J];安徽大學學報(自然科學版);2007年01期

8 張俊敏;成立花;李祚;;冪等矩陣線性組合的可逆性[J];純粹數(shù)學與應用數(shù)學;2007年02期

9 楊忠鵬;陳梅香;林國欽;;關(guān)于三冪等矩陣的秩特征的研究[J];數(shù)學研究;2008年03期

10 鄧春源;;廣義冪等算子差的可逆性[J];數(shù)學物理學報;2009年06期

相關(guān)博士學位論文 前2條

1 宋肖飛;與冪等陣相關(guān)的兩種保持問題[D];哈爾濱工業(yè)大學;2014年

2 生玉秋;矩陣空間的保持冪等關(guān)系的映射[D];哈爾濱工業(yè)大學;2007年

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