分配格上矩阵的特征向量的求解方法

发布时间：2018-02-27 23:02

本文关键词： 分配格完全完备分配格标准特征向量一般特征向量特征值　出处：《内蒙古工业大学》2015年硕士论文　论文类型：学位论文

【摘要】：从数学的系统结构来看,数学研究的对象分为:有序、代数、拓扑三个基本结构,格是兼有序和代数的重要结构,它和模糊数学、拓扑学等现代数学有十分紧密的联系;从格的概念出现于数学的各个领域可以看出,格对于数学及其他分支的联系及应用是非常重要的.在计算机科学、保密学、图论、泛函分析和开关理论等领域中,都直接应用了格.分配格是格论中非常重要的格,而矩阵是数学研究和数学应用的一个重要工具,因而在分配格矩阵上做研究是目前格论研究中比较热门的课题,在计算机理论、范畴论、拓扑代数和模糊数学中有着广泛的应用.本文主要研究分配格上矩阵的特征向量,从代数结构、性质和计算方法方面给出一些自己的见解和结论.文章主要分为三个部分:第一部分介绍了文章的研究背景、研究历史、研究现状,以及创新点;给出了全文所用到的一些定义、性质和引理.其中包括:格论部分给出了格的定义、格律以及格矩阵的运算和性质;图论部分介绍了伴随有向图、路径及预圈等相关知识;幂序列部分有幂序列、并既约元的定义及并既约元分解定理;完全完备分配格部分有伪补、完全完备分配格的定义及引理.第二部分首先介绍了两种方法求解分配格上矩阵的标准特征向量,第一种方法是用图论的知识求解分配格上矩阵的标准特征向量?x;第二种方法是用广义序列求解分配格上矩阵的标准特征向量,并举例说明.随后还讨论了分配格上对称矩阵的一般特征向量的求解方法.其次讨论分配格矩阵的全体标准特征向量的代数结构以及计算方法.为了后面研究特征向量的需要,首先介绍了分配格上矩阵的分解定理,指出分配格上矩阵可以有类似于格中元素的并既约分解,然后利用矩阵幂序列来给出分配格上矩阵的全部标准特征向量的计算公式,又进一步得出一般特征向量的求解方法.为求解标准特征向量,还给出了(n)A的计算方法.第三部分介绍了完全完备分配格上矩阵的特征向量的解法和性质.利用伪补给出了完全完备分配格上矩阵的最大特征向量的计算公式,并举例说明如何利用该公式求解最大特征向量;证明了完全完备分配格上矩阵的特征向量所对应的特征值构成一个区间,且给出区间端点的表达式,并指出如果特征向量有唯一特征值,该特征值所满足的范围.
[Abstract]:From the system structure of mathematics, the object of mathematical research is divided into three basic structures: order, algebra and topology. Lattice is an important structure of order and algebra, which is closely related to modern mathematics, such as fuzzy mathematics, topology and so on; From the concept of lattice appearing in various fields of mathematics, we can see that lattice is very important for the connection and application of mathematics and other branches, in the fields of computer science, secrecy, graph theory, functional analysis and switch theory, etc. Distribution lattice is a very important lattice in lattice theory, and matrix is an important tool in mathematical research and mathematical application, so doing research on distributive lattice matrix is a hot topic in lattice theory research. It is widely used in computer theory, category theory, topological algebra and fuzzy mathematics. The article is divided into three parts: the first part introduces the research background, the research history, the present research situation, and the innovation; Some definitions, properties and Lemma used in this paper are given, including the definition of lattice, lattice law, operation and properties of lattice matrix in the lattice theory part, the associated digraph, path, precycle and other related knowledge in graph theory. The part of power sequence has power sequence, and the definition of the reduction element and the decomposition theorem of the irreducible element, the complete complete distributive lattice part has pseudo complement, The definition and Lemma of complete Distributive Lattice. In the second part, we introduce two methods to solve the standard eigenvector of matrix on distributive lattice. The first method is to solve the standard eigenvector of matrix on distributive lattice by using the knowledge of graph theory. The second method is to use generalized sequences to solve the standard eigenvector of matrices on distributive lattices. Then, the method of solving the general eigenvector of symmetric matrix on distributive lattice is discussed. Secondly, the algebraic structure and calculation method of all standard eigenvector of distributive lattice matrix are discussed. This paper first introduces the decomposition theorem of matrices on distributive lattices, points out that matrices on distributive lattices can have paratactic factorization similar to elements in a lattice, and then gives the calculation formulas of all standard eigenvectors of matrices on distributive lattices by using matrix power sequence. In order to solve the standard eigenvector, In the third part, the solution and properties of the eigenvector of the matrix on the complete complete distributive lattice are introduced. The formula of the maximum eigenvector of the matrix on the complete complete distributive lattice is obtained by using pseudo resupply. An example is given to illustrate how to use the formula to solve the maximum eigenvector, it is proved that the eigenvalues corresponding to the eigenvector of the matrix on the complete complete distribution lattice form an interval, and the expression of the end point of the interval is given. It is pointed out that if the eigenvector has a unique eigenvalue, the range of the eigenvalue can be satisfied.
【学位授予单位】：内蒙古工业大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O153.1

【共引文献】