非奇异不可约M矩阵Hadamard积的最小特值下界估计

发布时间：2018-06-16 22:58

本文选题：非负矩阵 + M矩阵　；参考：《太原理工大学》2017年硕士论文

【摘要】：M矩阵是计算数学学科研究中的主要分支,常用来解决物理学,经济学和生物学等方面的问题,而M矩阵的最小特征值下界估计是矩阵理论中主要概念之一,故有重要的研究意义。论文以现有文献为基础,利用Gersgorin圆盘定理,给出了非奇异不可约M矩阵A和双随机矩阵-1A的Hadamard积的最小特征值下界估计式,利用矩阵特征值存在域定理,给出了两个非奇异不可约M矩阵A和B的Hadamard积的最小特征值下界估计式。本文结构组织如下:第一章,对非负矩阵,M矩阵,矩阵Hadamard积的产生及应用背景和国内外研究现状进行了介绍,并对本文的研究成果也做了介绍。第二章,首先介绍了非负矩阵,不可约矩阵,M矩阵,以及Hadamard积,谱半径等方面的基础知识,其次介绍了本文要用到的一些已有的结论,引理和定理。第三章,论文研究的主要成果之一,利用Gersgorin圆盘定理,对非奇异不可约M矩阵A和双随机矩阵A~(-1)进行研究,给出了Hadamard积AAo~(-1)两个新的最小特征值（?）下界估计式及证明。估计式如下:和并证明了该估计式比现有文献的结果要好,且通过数值算例表明所得到的估计式比现有文献的估计式更加精确,并且估计式只与矩阵元素相关,易于计算。第四章,论文研究的主要成果之二,在现有文献的基础上,给出了一个非奇异不可约M矩阵B和另一个非奇异不可约M矩阵A的逆矩阵的Hadamard积(?)的最小特征值下界估计式及证明。估计式如下:通过数值算例表明所得到的估计式比现有文献的估计式更加精确。总结与展望,总结了本文所做的研究工作,并提出了文中的不足之处和值得继续研究的方向。
[Abstract]:M matrix is a major branch of computational mathematics, which is often used to solve physics, economics and biology problems. The minimum eigenvalue lower bound estimation of M matrix is one of the main concepts in matrix theory. Therefore, it has important research significance. Based on the existing literatures and using Gersgorin's disk theorem, the paper gives the minimum eigenvalue lower bound estimator of the Hadamard product of nonsingular irreducible M matrix A and double random matrix -1A, and uses the existence domain theorem of matrix eigenvalue. The lower bound estimators of the minimum eigenvalue of Hadamard product of two nonsingular irreducible M matrices A and B are given. The structure of this paper is as follows: in Chapter 1, the production and application background of non-negative matrix M matrix, matrix Hadamard product and the research status at home and abroad are introduced, and the research results of this paper are also introduced. In chapter 2, we first introduce the basic knowledge of nonnegative matrix, irreducible matrix M matrix, Hadamard product and spectral radius, and then introduce some existing conclusions, Lemma and theorems which will be used in this paper. In chapter 3, one of the main achievements of this paper, using Gersgorin's disk theorem, we study the nonsingular irreducible M matrix A and the double random matrix A ~ (1), and give two new minimum eigenvalues of Hadamard product AAoM-1). Lower bound estimate and proof. The estimators are as follows: and it is proved that the estimator is better than the existing ones, and the numerical examples show that the obtained estimators are more accurate than the existing ones, and the estimators are only related to matrix elements and are easy to calculate. In the fourth chapter, one of the main achievements of this paper, the Hadamard product of the inverse matrix of a nonsingular irreducible M matrix B and another nonsingular irreducible M matrix A is given on the basis of the existing literatures. The estimation and proof of the lower bound of the minimum eigenvalue of this paper. The estimation formula is as follows: the numerical example shows that the obtained estimator is more accurate than the existing literature. This paper summarizes the research work done in this paper, and puts forward the deficiency and the direction of further study.
【学位授予单位】：太原理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.6

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