关于Loewner矩阵的若干问题-应用数学专业论文.docxVIP

2 2009 年 4 月 安徽大学硕士学位论文 (邹奎方) i i 2 2009 年 4 月 安徽大学硕士学位论文 (邹奎方) 2 2009 年 4 月 安徽大学硕士学位论文 (邹奎方) i i ii ii 摘 要 在现代线性代数中，Loewner 矩阵以及各种推广有着非常重要的应用，这引 起学者们的广泛重视，并得出了很多重要的成果.最近，Loewner 矩阵更多的是 与 Hankel 矩阵、Bezout 矩阵联系在一起，且在有理插值理论中，Loewner 矩阵 以及其各种推广形式起着基础性的作用.因此，研究 Loewner 矩阵具有重要的现 实意义. 本文就经典 Loewner 矩阵、广义 Loewner 矩阵以及它们与 Bezout 矩阵、 Hankel 矩阵之间的关系作了一些研究，本硕士论文共分三章： 第一章叙述了问题产生的背景与意义及本文所做的主要工作； 第二章讨论了经典（对称）Loewner 矩阵的相关概念与性质，以及它们与 Hankel 矩阵、Bezout 矩阵之间的关系； 第三章讨论了广义 Loewner 矩阵的相关概念，并在第二章的基础上把有些结 果进行了推广，最后证明了 Loewner 矩阵空间与 Hankel 矩阵空间的一种同构关 系. 关键词 Loewner 矩阵；广义 Loewner 矩阵；Hankel 矩阵；Bezout 矩阵； ABSTRACT The concept of Loewner matrix and its various generalization play an important role in modern linear algebra. And this phenomenon has attacted many scholars’ attention and lots of available and important results have been obtained. More recently Loewner matrix appeared in connection with Hankel and Bezout matrix and in the theory of the rational interpolation ,Loewner matrix and its various generalization play an essential role，so it is important practical meanings to study Loewner matrix. In this paper, the classical Loewner and generalized Loewner matrices,Hankel matrices ,Bezoutians and the relation between them are discussed .This thesis is composed of three chapters: In chapter one, the backgrounds and significance of the problems are given .The main work done is introduced. In chapter two we mainly consider the classical Loewner and symmetrical Loewner matrices as well as Bezout and Hankel matrices. The relation between them are discussed. In chapter three, we consider generalized Loewner matrices and generalize the results discussed in chapter two .Finally we prove the the isomorphism relation of the Hankel matrices space and Loewner matrices space. Key Words Loewner matrix; generalized Loewner matrix; Bezout matrix; Hankel matrix. iii iii 目录 摘要…………………………………………………………….ⅰ Abstract……………………………………………………….ⅱ 第一章 绪论……………………………………………………1 第二章 Loewner 矩阵………………………………………….4 §2.1 引言