非奇异H-矩阵的判定和张量谱半径的估计

发布时间：2018-03-06 12:36

本文选题：非奇异H-矩阵　切入点：对角占优矩阵　出处：《吉林大学》2016年博士论文　论文类型：学位论文

【摘要】：非奇异H-矩阵是一类特殊却又极为重要的矩阵,它在许多领域都有着不容忽视的作用,例如：矩阵理论、数量经济学、概率统计、控制论、电力系统理论、神经网络以及数学物理和社会科学等.然而,如何判定一个给定的矩阵是否为非奇异H-矩阵是基本而又困难的问题.近年来,国内外许多学者已经在这方面开展了研究并给出了非奇异H-矩阵的一些有效的判别方法.本文在已有工作的基础上,利用具有非零元素链矩阵和不可约矩阵等的相关理论知识,采用寻找正对角矩阵因子的方法,并借助于不等式的放缩技巧给出了判定非奇异H-矩阵更广泛、更有效的准则,改进了已有的一些结果,并通过一些例子说明了所得结果的有效性和实用性.另一方面,张量作为矩阵的高阶推广,它仍保留着矩阵的很多性质.同时,张量在诸如盲源分离、磁共振成像、非线性优化和高阶统计学等很多领域都有着广泛的应用.因此,本文中我们也将对张量进行研究,主要考虑张量的谱半径估计,通过研究张量的特征值问题,给出了非负张量谱半径的一个新的界,证明所给出的界优于文献中的已有结果,并通过具体例子说明了所得结果的有效性和实用性.
[Abstract]:Non-singular H-matrix is a kind of special but extremely important matrix. It plays an important role in many fields, such as matrix theory, quantitative economics, probability statistics, cybernetics, power system theory, etc. However, how to determine whether a given matrix is a nonsingular H-matrix is a basic and difficult problem. Many scholars at home and abroad have carried out research in this field and given some effective discriminant methods for nonsingular H-matrices. On the basis of existing work, this paper makes use of the relevant theoretical knowledge of nonzero element chain matrix and irreducible matrix, etc. By using the method of finding positive diagonal matrix factors, and by means of the scaling technique of inequality, a more extensive and effective criterion for judging nonsingular H-matrices is given, and some results are improved. On the other hand, Zhang Liang, as a high order generalization of the matrix, still retains many properties of the matrix. At the same time, Zhang Liang in such as blind source separation, magnetic resonance imaging, Nonlinear optimization and high order statistics are widely used in many fields. Therefore, we will also study Zhang Liang in this paper, mainly considering the spectral radius estimation of Zhang Liang, and studying the eigenvalue problem of Zhang Liang. A new bound of the spectral radius of non-negative Zhang Liang is given. It is proved that the given bound is superior to the existing results in the literature, and the validity and practicability of the obtained results are illustrated by concrete examples.
【学位授予单位】：吉林大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O151.21

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