Ray非异矩阵的禁用结构

The concept of ray pattern matrices is a natural generalization of that of sign pattern matrices in the complex field. Correspondingly, ray non-singular matrices are a natural generalization of the sign non-singular matrices (abbreviated as SNS matrices). The SNS matrices whose diagonal entries are all negative were characterized by the forbidden structure positive cycles. In 2013, a type of forbidden structure of ray non-singular matrices was found, which was called forbidden cycle chains. This project is willing to take the research of the forbidden structures of ray non-singular matrices into the further step. Specifically it contains the following three parts: 1. whether or not there exist some minimum forbidden structures of ray non-singularity other than forbidden cycle chains; 2. the probability of the emergence of the forbidden structures in some probability spaces; 3. the spectral arbitrariness and other related properties of the ray pattern matrices those contain some forbidden structures.

Ray模式矩阵是符号模式矩阵在复数域下的一种自然推广。相应地，Ray非异矩阵是符号非异矩阵（SNS矩阵）的一种自然推广。对角元全负的SNS矩阵是应用其伴随有向图禁用结构——正圈——来进行刻画的。2013年，我们发现了Ray非异矩阵的一种禁用结构，称为禁用圈链。本项目拟进一步研究Ray非异矩阵的禁用结构。主要内容包括：1. 研究除禁用圈链外是否还存在其它的Ray非异矩阵的极小禁用结构；2. 讨论在一些特定概率空间中禁用结构出现的概率；3. 研究含有禁用结构的Ray模式矩阵的谱任意性等其它相关组合性质。

本项目主要研究复矩阵的组合结构与其代数性质之间的联系，是非负矩阵和符号矩阵理论的推广。本项目主要研究了复方阵Ray非异性的禁用结构。我们证明了若不含正圈，则圈树矩阵M是Ray非异性的禁用结构当且仅当M不是Ray非异矩阵；我们引入了复平面单位圆周上均匀分布随机变量的定义，并证明了其具有卷积不变性。我们给出了在相应概率模型下Ray非异性的概率函数或阈值函数；我们证明了Ray模式矩阵蕴含Perron-Frobenius性的充分必要条件是所有圈都是正圈。此外我们还找到了两种新的谱任意Ray模式矩阵。