【作者】 刘月；

【导师】 邵嘉裕；

【作者基本信息】 同济大学 ， 应用数学， 2006， 硕士

【摘要】 符号矩阵理论是组合矩阵论的一个新兴研究分支，是近年来在组合数学中较为活跃的一个研究方向。该理论主要研究矩阵的仅与其符号模式有关的定性性质。符号矩阵理论最早起源于经济学中对某些问题的定性性质的研究。其开创性工作是由诺贝尔奖获得者、经济学家P．Samuelson作出的(参见文献[16])。由于符号矩阵理论在经济学中有着重要的应用背景，从而引起了经济学家，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的关于符号矩阵论的专著《Matrices of Sign-solvable Linear Systems》([5]的问世极大地推动了符号矩阵理论的发展，它全面系统地总结了在符号矩阵理论方面的研究成果，同时给出了许多新的结论，从而使符号矩阵理论成为组合数学的一个新兴的研究热点。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,D.D.Olesky,M.J.Tsatsomeros和P.van den Driessche在文献[14]中将符号模式矩阵的概念推广到ray模式矩阵，并研究了ray模式矩阵的ray非奇异性及ray模式矩阵类的行列式值域的问题。1998年，C.A.Eschenbach,F.J.Hall,and Z.Li.等在文献[6]中从另一个方面对符号模式矩阵的概念做了推广，即所谓的“复符号模式矩阵”。在以上两种推广下都有一个基本问题有待解决，即ray非异阵或复符号非异阵的特征刻画。为了便于研究这个问题，在文献[14]中McDonald等同时提出了行列式值域的概念。2005年，在文献[20]中Jia-Yu Shao和Hai-YingShah对行列式值域问题进行了深入的研究，给出了一些行列式值域的必要条件，并据此列出了所有可能成为某一个矩阵在复符号模式下的行列式值域的区域形状。对于ray模式矩阵的行列式值域，他们定义了一个和行列式值域的形状密切相关的参数：行列式值域的叶数，并关于叶数这个参数提出了如下一些问题。问题1：叶数是否一定有限?问题2：如果叶数有上界，确定叶数的上确界，同时给出不同叶数的行列式值域的特征。对于复符号模式可以提出类似的问题。本文将主要对叶数这个参数进行研究。在文中我们定义了一种矩阵间的距离，利用这个概念我们证明复方矩阵在ray模式下行列式值域的叶数一定有限，并且最大可能值是2，然后我们证明叶数为2的矩阵行列式值域一定关于原点对称。利用上述结论我们列出了所有可能的ray模式下的行列式值域的形状，共9类，其中还有1类没有完全确定。我们在ray模式下的证明在复符号模式下也是可行的，利用复符号模式下相应的结论，我们排除了在文献[20]未被确定的6类可能成为行列式值域的区域中的4类，剩余2类共8种可能的区域还没有确定。更多还原

【Abstract】 Signed matrix theory is a new branch of combinatorial ma-trix theory, which involves the study of the properties of matricesthat depend only on the sign pattern of the matrices. The sub-ject, started by the Nobelist P.Samuelson who is an economist, isusually considered to have originated with the discussion on "qual-itative properties" of some problems in economics (see [16]). Be-cause of its great importance in economics, it has been paid wideattention by many researchers including economists, mathemati-cians and theoretical computer scientists. In 1995, R.A.Brualdi andB.L.Shader’s book《Matrices of sign-solvable linear systems》([5])was published, which is the first monograph on signed matrix the-ory. In this book, they systematically summarized the results onthe subject and gave many new results. It makes signed matrixtheory being an active area of combinatorics.Recently, the research of signed matrix theory transfers fromreal field to complex field. In 1997, J.J.McDonald etc. generalizedthe concept of the qualitative class to ray pattern class, and stud-ied questions on the nonsingularity and determinantal regions of raypattern matrices. In 1998, Eschenbach etc. extended the conceptof the qualitative class in another way in [6], so called the complexsign pattern. In both ways of extension, a fundamental questionis out there: what is the characterization of nonsingular matricesin ray pattern or in complex sign pattern? In order to investigatethis problem, the definition of determinantal regions of matrices isreferred in [14]. In 2005, J.Y.Shao and H.Y.Shan did a deep re-search in determinantal regions of matrices in [20]. Some necessaryconditions of determinantal regions is given and all possible regionswere listed (22 types altogether 74 possible regions, 6 types were leftundetermined). Also an important parameter denoted by n_R(A) isreferred, and some questions were asked.Question 1: Is n_R(A) always finite when A ranges over all com-plex square matrices?Question 2: If the answer to Question 1 is true, determine themaximal possible value of n_R(A), and characterize cases with dif-ferent n_R(A). Similar questions may be raised for the complex sign pattern.We mainly discussed the parameter n_R(A) in this paper. Firsta kind of distance between two matrices with the same size is de-fined here. Then it is proved that the maximal possible value ofn_R(A) is 2, further more the characterization of case n_R(A)=2is given. After that all possible regions as determinantal regionsof ray pattern matrices are listed, there are 9 types and only onetype is left undetermined. The proof is also effective for complexsign pattern. So use these conclusions, 4 of the 6 types left in[20] are excluded for determinantal regions of complex sign patternmatrices.更多还原