极大代数上非负区间矩阵的谱

发布时间：2018-06-08 17:07

本文选题：极大代数 + 非负区间矩阵　；参考：《河北师范大学》2017年硕士论文

【摘要】：极大代数是研究通讯网络、交通控制、灵活制造等离散事件系统的最常用的工具之一.实际问题所涉及的时间未必是定值,更多的情形是受到某些外在或内在因素的影响在一个范围内变化.因此,以区间作为元素的极大代数为解决这些实际问题提供了一种有效的代数方法,而对于此类极大代数,矩阵谱的计算是重要的研究内容之一.本文主要从以下几部分来研究极大代数上非负区间矩阵的谱.首先,给出极大代数意义下非负区间和非负区间矩阵的定义及相关运算,并定义非负区间上的偏序关系,进而给出了极大回路几何平均区间、谱、非负区间矩阵的有向图及简化图等概念.其次,通过定义非负区间向量的范区间和局部谱半径,研究了极大回路几何平均区间与局部谱半径之间的关系,借助极大锥,得到了谱的一种数学表达形式;进而利用有向图中的可达关系,给出了谱的另一种数学表达形式.与此同时,给出了任意非负区间矩阵特征值的简便算法.再次,定义了任意两个非负区间矩阵的克罗内克积,研究了其特征值与这两个非负区间矩阵的特征值之间的关系,从而得到克罗内克积的谱的表达.最后,在极大代数意义下,定义非负区间多项式集,并验证谱映射定理.
[Abstract]:Maximal algebra is one of the most commonly used tools to study discrete event systems such as communication networks, traffic control and flexible manufacturing. The time involved in a practical problem is not necessarily a fixed value, but more often changes within a range under the influence of some external or internal factors. Therefore, the maximal algebra with interval as the element provides an effective algebraic method for solving these practical problems. For this kind of maximal algebra, the calculation of matrix spectrum is one of the important research contents. In this paper, the spectrum of nonnegative interval matrices on maximal algebra is studied in the following parts. Firstly, the definitions of nonnegative interval matrix and nonnegative interval matrix in the sense of maximal algebra and their correlation operations are given, and the partial order relation on the nonnegative interval is defined, and then the geometric mean interval and spectrum of maximum loop are given. The concept of directed graph and simplified graph of nonnegative interval matrix. Secondly, by defining the normed interval and the local spectral radius of the non-negative interval vector, the relationship between the geometric mean interval of the maximum loop and the local spectral radius is studied, and a mathematical expression of the spectrum is obtained with the help of the maximal cone. Then another mathematical expression of spectrum is given by using the reachability relation in digraph. At the same time, a simple algorithm for eigenvalues of arbitrary nonnegative interval matrices is given. Thirdly, the Cronecker product of any two nonnegative interval matrices is defined, and the relationship between the eigenvalues and the eigenvalues of these two nonnegative interval matrices is studied, and the spectral representation of the Cronecker product is obtained. Finally, in the sense of maximal algebra, the set of nonnegative interval polynomials is defined and the spectral mapping theorem is proved.
【学位授予单位】：河北师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O151.21

【相似文献】