极大加代数矩阵的整特征向量和整像

发布时间：2018-05-24 07:43

本文选题：算法 + 特征值　；参考：《河北师范大学》2015年硕士论文

【摘要】：极大加代数是研究系统科学的一个重要工具,它可使时序离散事件系统具有像一般线性系统那样的状态空间表达式,达到利用线性系统模型分析离散事件动态系统的目的.极大加代数能解决许多实际问题,例如,资源分配问题,铁路系统调度问题,生产流水线最优控制问题,制造系统优化和控制问题等.在极大加代数结构中重新定义了许多重要的数学概念,并讨论其性质,例如,极大加代数矩阵的行列式及其性质、极大加代数矩阵的特征值和特征向量、线性独立、模结构等.矩阵的特征问题和像问题是极大加代数中重要的数学概念且有着重要的理论和实践意义.特征值表示系统的周期时间,特征向量表示系统的稳定状态.像表示系统的运行结果.本文的主要内容是在前人工作的基础上,研究极大加代数矩阵的整特征向量和整像.给出可约极大加代数矩阵存在块整特征向量的充分必要条件和在一定条件下存在整特征向量的充分必要条件.给出广义整像算法,数值例子表明广义整像算法是伪多项式算法.给出??×3极大加代数矩阵和Monge矩阵存在整像的充分必要条件,并且给出强确定矩阵存在整特征向量和存在整像的等价性.本文共分为五个部分.引言部分,介绍与极大加代数矩阵整特征向量和整像相关的研究背景和研究现状.第一章,介绍本文涉及到的基本概念和引理,包括极大加代数、可约极大加代数矩阵、特征值、特征向量、整像等.举例说明极大加代数矩阵间的运算.这些概念和引理为后面的章节提供了理论支撑.第二章,介绍极大加代数矩阵的整特征向量.提出块整特征向量的概念,分别给出可约极大加代数矩阵存在块整特征向量的充分必要条件和在一定条件下存在整特征向量的充分必要条件并给出相应的数值例子.第三章,介绍极大加代数矩阵的整像.给出??×3极大加代数矩阵存在整像的充分必要条件,??×??极大加代数矩阵特殊条件下存在整像的充分必要条件.给出广义整像算法,通过验证主对角线上的块极大加代数矩阵确定可约和不可约极大加代数矩阵的整特征向量.数值例子表明广义整像算法是伪多项式算法.最后给出Monge矩阵存在整像的充分必要条件和强确定矩阵存在整特征向量和存在整像的等价性.结论部分,总结本篇论文的主要结论,并提出有待进一步研究的问题.
[Abstract]:The maximum additive algebra is an important tool for the study of system science. It can make the time series discrete event system have the state space expression like the general linear system, and achieve the purpose of analyzing the discrete event dynamic system by using the linear system model. The problem of unified scheduling, optimal control of production lines, optimization and control of manufacturing systems, and so on. Many important mathematical concepts are redefined in the maximal additive algebraic structure, and its properties are discussed, such as the determinant and its properties of the maximal additive algebraic matrix, the eigenvalues and eigenvectors of the maximal additive number matrix, linear independence, and the model knot. The characteristic problems and image problems of matrices are important mathematical concepts in maximal addition algebra and have important theoretical and practical significance. The eigenvalues represent the periodic time of the system, the eigenvectors represent the stable state of the system. The full and necessary conditions for the existence of integral eigenvectors of an algebraic matrix with an algebraic matrix and the sufficient and necessary conditions for the existence of an integral eigenvector under certain conditions are given. The generalized integer image algorithm is given, and the numerical example shows that the generalized integer image algorithm is a pseudo multi term algorithm. The?? * 3 maximal additive algebraic matrix is given. The sufficient and necessary conditions for the existence of an integer image with the Monge matrix and the equivalence of the existence of an integer eigenvector and the existence of an integer image are given. This paper is divided into five parts. The introduction part introduces the research background and research status related to the integral eigenvector and integer image of the maximal additive algebra. Chapter 1 introduces the basic probability of this article. The idea and lemma, including the maximal addition of algebra, can be greatly added to the algebraic matrix, eigenvalues, eigenvectors, and integer images. Examples are given to illustrate the operations between the maximal added algebraic matrices. These concepts and lemmas provide theoretical support for the subsequent chapters. The second chapter introduces the entire eigenvector of the maximal additive algebraic matrix. The sufficient and necessary conditions for the existence of integral eigenvectors of the approximately maximal additive algebraic matrices and the sufficient and necessary conditions for the existence of an integral eigenvector under certain conditions are given and the corresponding numerical examples are given. In the third chapter, the entire image of the maximal additive algebraic matrix is introduced. The generalized integer image algorithm is given. The generalized integer image algorithm is given, and the integral eigenvector of the reducible and irreducible algebraic matrix is determined by verifying the maximal addition algebraic matrix of the block on the main diagonal line. The numerical example shows that the generalized integer image algorithm is a pseudo multi term algorithm. Finally, the existence of the Monge matrix is given. The sufficient and necessary conditions of the image and the equivalence of the integral eigenvector and the existence of the whole image exist in the sufficient and necessary condition of the image and the strong deterministic matrix.
【学位授予单位】：河北师范大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O151.21

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