一类X型矩阵特征值反问题

发布时间：2018-01-16 11:38

本文关键词：一类X型矩阵特征值反问题　出处：《大连交通大学》2015年硕士论文　论文类型：学位论文

【摘要】：矩阵特征值反问题是线性代数的一个重要分支,在跨越广泛的科学领域中应用普遍。自20世纪50年代第一篇关于这方面文章发表以来,越来越多的此方面研究论文相继公开发表,获得了很多深刻而且有益的结果。当今研究的目标就是构造出了一些实际科学应用中需要的特征向量和特征值的矩阵。本文研究目标定位于构造一类X型矩阵,从而研究其特征值反问题及其广义特征值反问题,利用方程组联立求解并递推得出问题解存在并且唯一的条件。全文由以下四章构成,内容如下:第一章:绪论。首先介绍反问题概念、历史,其次介绍矩阵特征值反问题的当今研究现状、难点及未来应用,最后对本论文所研究的X型矩阵的相关概念进行介绍。第二章:一类基本X型矩阵特征值反问题。本章首先提出一类X型矩阵的特征值反问题,并对矩阵存在的条件进行推导,得出一类X型矩阵特征值反问题解存在并且唯一所需要满足的条件,在此基础上将X型矩阵右上角的元素剔除,从而得到了我们常见的下三角矩阵,也可称其为退化X型矩阵,按照研究X型矩阵特征值反问题的方法,对此类下三角矩阵的特征值反问题进行研究并得到一类退化X型矩阵特征值反问题解存在并且唯一所需要满足的条件并给出解的表达式。第三章:一类特殊退化X型矩阵的特征值反问题。本章在第二章中提出的退化X型矩阵的基础上加以改动,得到一类上三角矩阵,并分别将矩阵元素之间的关系按照等值关系、线性关系分为两类,然后对每一类矩阵的特征值反问题进行研究,分别得到一类特殊退化X型矩阵特征值反问题解存在并且唯一所需要满足的条件并给出解的表达式。最后给出两个相应的数值例子分别进行了验证。第四章:一类特殊退化X型矩阵的广义特征值反问题。本章在前三章的基础上,将对矩阵特征值反问题的研究扩展到对矩阵广义特征值反问题的研究上,研究了一类奇数阶上三角矩阵的广义特征值反问题,得出一类特殊退化X型矩阵广义特征值反问题解存在并且唯一的条件并给出解的表达式。最后给出数值例子对算法的有效性进行验证。
[Abstract]:Inverse eigenvalue problem of matrices is an important branch of linear algebra, which is widely used in many fields of science. Since 1950s, the first article on this field has been published. More and more research papers have been published in this field. The goal of the present study is to construct some characteristic vectors and eigenvalues needed in practical scientific applications. The purpose of this paper is to construct a class of X-type matrices. Therefore, the inverse eigenvalue problem and its generalized inverse eigenvalue problem are studied, and the existence and unique conditions of the solution are obtained by simultaneous solution of equations. The paper is composed of four chapters. The contents are as follows: chapter one: introduction. Firstly, the concept of inverse problem, history, and then the current research status, difficulties and future application of inverse matrix eigenvalue problem are introduced. Finally, the related concepts of X-type matrix studied in this paper are introduced. Chapter two: the inverse eigenvalue problem of a basic X-type matrix. In this chapter, we first propose a class of inverse eigenvalue problem of X-type matrix. The condition of the existence of matrix is deduced, and the condition that the inverse solution of eigenvalue of type X matrix exists and only needs to be satisfied is obtained. On this basis, the elements in the upper right corner of the matrix of type X are eliminated. Thus we get our common lower triangular matrix, which can also be called degenerate X-type matrix, according to the method of studying inverse eigenvalue problem of X-type matrix. In this paper, we study the inverse eigenvalue problem of this kind of lower triangular matrices, and obtain the conditions for the existence and uniqueness of the inverse eigenvalue problem of a class of degenerate X-type matrices, and give the expression of the solution. A class of inverse eigenvalue problems for a class of special degenerate X-type matrices. This chapter is modified on the basis of the degenerate X-type matrices proposed in Chapter 2. A class of upper triangular matrices is obtained, and the relations between matrix elements are divided into two categories according to the equivalence relationship. Then the inverse eigenvalue problem of each kind of matrix is studied. The conditions for the existence and uniqueness of inverse solutions of eigenvalues of a special degenerate X-type matrix are obtained and the expressions of the solutions are given. Finally, two corresponding numerical examples are given. Chapter 4th:. Generalized eigenvalue inverse problem for a class of special degenerate X-type matrices. This chapter is based on the previous three chapters. In this paper, the inverse problem of matrix eigenvalue is extended to the inverse problem of generalized eigenvalue of matrix, and the inverse problem of generalized eigenvalue of a class of upper triangular matrices of odd order is studied. The existence and uniqueness conditions of inverse solutions for a class of special degenerate X-type matrices with generalized eigenvalue problems are obtained and the expressions of the solutions are given. Finally, a numerical example is given to verify the validity of the algorithm.
【学位授予单位】：大连交通大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O151.21

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