孤立置换集，复方阵的行列式值域和完全拟S～*阵

【作者】 任灵枝；

【导师】 邵嘉裕；

【作者基本信息】 同济大学 ， 应用数学， 2006， 博士

【摘要】 作为组合矩阵论的一个分支，符号矩阵论主要研究矩阵的定性性质，即仅与矩阵的符号模式有关的性质．它最早始于诺贝尔奖获得者、经济学家P．Samuelson在1947年所开创的关于符号可解的线性方程组的研究([30])，此后，因其在定性经济学中的重要应用及作为线性代数、组合数学和图论的交叉学科，符号矩阵理论引起了经济学家、数学家及计算机理论专家的广泛关注，1995年，R．A．Brualdi与B．L．Shader在符号矩阵论的第一部专著《Matrices of Sign-solvable Linear Systems》([9])中总结并极大丰富了符号矩阵理论的研究成果，从而使符号矩阵论成为组合矩阵论的一个研究热点．近年来，符号矩阵理论的研究有着从实数域向复数域推广的趋势，1997年，J．J．McDonald等人在[28]中将实符号非异阵推广为ray符号非异阵，他们在研究ray符号模式矩阵的行列式值域时，引入了孤立的transversal集合的概念，并给出一个阶数至多为3的transversal集合为孤立的时的刻划，我们在第二章中，说明了孤立的transversal集合等价于孤立的置换集(§2．1)，并给出了孤立的置换集的图论刻划(§2．2)，此外，我们利用孤立置换集的刻划解决了一个关于某些置换阵的线性无关性判定的问题(§2．3)．1998年，C．A．Eschenbach等人在[11]中将实符号模式矩阵推广为复符号模式矩阵，并提出了若干值得研究的问题，其中之一是“给出复符号模式矩阵所有可能的行列式值域”，J．Y．Shao和H．Y．Shan在[41]中列出了复符号模式矩阵的行列式值域可能的区域类型，除这些类型的区域以外，他们猜想剩余类型的区域一(B1)-(B11)均不可能为任一复符号模式矩阵的行列式值域，其中的(B1)-(B3)，(B6)-(B7)类型的区域在[41]中就已经被排除，本文在第三章中排除了(B8)型区域(§3．2)，以及(B4)，(B9)，(B10)型区域(§3．3)。完全拟S~*阵是符号矩阵论中重要的一类矩阵，[9]对之作过专门的论述，[9]中叙述了完全拟S~*阵的零位模式的刻划，本文在第四章中独立地给出了完全拟S~*阵的符号模式的刻划(§4．1)，并通过定义完全拟S~*带号二分图和研究完全拟S~*带号二分图的结构给出了完全拟S~*阵的结构(§4．2)。Nearly拟S~*阵是与拟S~*阵相关的一类矩阵，本文在第五章中刻划了nearly拟S~*阵(§5．1)，并通过定义nearly拟S~*有向二分图和研究nearly拟S~*有向二分图的结构给出了所有的nearly拟S~*阵(§5．2)。更多还原

【Abstract】 As a branch of combinatorial matrix theory, signed matrix theory mainlystudies a matrix, s qualitative properties which only depend on the matrix, s signpattern.The subject originated from the discussion about the sign-solvable linearsystems which was started in 1947 by P. Samuelson, a Nobelist and an economist.Since then, signed matrix theory has attracted much attention from economists, mathematicians and computer scientists due to its important economic applica-tions and the beautiful interplay it afforded among linear algebra, combinatoricsand theoretical computer science. In 1995, the first book on signed matrix theory-《Matrices of Sign-solvable Linear Systems》([9])by R.A.Brualdi and B.L.Shadersummarized systematically the previous research results and also gave many newresults, which made signed matrix theory an active research field in combinatorics.Recently, some scholars have done a job which aims at generalizing the resultsin signed matrix theory from real field to complex field. In 1997, J.J.McDonaldetal. introduced ray nonsingular matrices as a generalization of real nonsingularmatrices in[28]. During the course of studying the determinantal regions of raypattern matrices, they advanced the concept of isolated transversal set, and theyalso gave a characterization of an isolated transversal set whose size is at most3. In Chapter2, we point out that an isolated transversal set is equivalent toan isolated permutation set(§2.1), and give graph theoretical characterizationsof an isolated permutation set(§2.2). In addition, we have solved a problem onjustifying the linear independence of some permutation matrices as an applicationof the characterizations of an isolated permutation set(§2.3).In 1998, C.A.Eschenbach etal. generalized real sign pattern matrices to com-plex sign pattern matrices in [11], and put up with some important open problems, one of which is "Giving all the possible regions of complex sign pattern matrices".In [41], J.Y.Shao and H.Y.Shan listed some possible regions of complex sign pat-tern matrices, in addition to these regions, they conjectured that all of the restregions—(B1)-(B11) can’t be a determinantal region of any complex sign patternmatrix. (B1)-(B3) and (B6)-(B7) have been excluded in [41]. In Chapter3, weexclude (B8)(§3.2), and (B4)、(B9)、(B10)(§3.3).Totally quasi S~* matrices are an important matrix class on which [9]did aspecial discussion. A characterization on zero patterns of totally quasi S~* matriceswas given in [9]. In Chapter4, we independently obtain a characterization of signpatterns of totally quasi S~* matrices(§4.1), and give the structure of totally quasiS~* matrices after defining totally quasi S~* signed bipartite graphs and studyingthe structure of totally quasi S~* signed bipartite graphs(§4.2).Nearly quasi S~* matrices are a class of matrices which has a close relation-ship with quasi S~* matrices. In Chapter5, we characterize nearly quasi S~* ma-trices(§5.1), and give all of nearly quasi S~* matrices after defining nearly quasi S~* bipartite digraphs and studying the structure of nearly quasi S~* bipartite di-graphs(§5.2).更多还原