矩阵奇异值及酉不变范数的研究

发布时间：2018-01-12 18:36

本文关键词：矩阵奇异值及酉不变范数的研究　出处：《陕西师范大学》2015年硕士论文　论文类型：学位论文

【摘要】：矩阵理论不但具有丰富的研究内容,又是一门最有实用价值的数学工具.在矩阵理论中,奇异值分解和奇异值不等式在高水平的统计计算和酉不变范数理论等中起着重要的作用.多年来,许多专家学者从不同的角度用不同的方法对有关奇异值的问题进行了很多的研究和推广,拓展了奇异值的应用范围和理论体系.本文用不同的方法对奇异值分解和Weyl定理进行了证明,并给出了一些推论.同时酉不变范数作为矩阵理论的一个重要数值特征,在矩阵计算、最佳逼近问题及扰动理论中有重要的作用.此外,分块矩阵作为一种特殊的矩阵,是探讨矩阵内在性质的重要工具,本文将半正定2×2分块矩阵和酉不变范数结合起来,得到了一系列矩阵的酉不变范数不等式.本文共分三章,具体内容如下：第一章主要介绍了本文中涉及的一些符号概念,主要有正规矩阵、酉矩阵、Hadamard积、弱受控、log-弱受控、压缩矩阵、半正定矩阵的半正定平方根、矩阵的广义逆,同时又介绍了一些已知的结论和定理.第二章首先给出了奇异值分解定理,然后分别从谱分解和极分解的角度给出了奇异值分解的其它证明方法,并由此得出了主对角元向量和奇异值向量的弱受控关系,并给出了Weyl定理的其它证明及推论.第三章利用酉不变范数和分块矩阵的一些性质,将酉不变范数和半正定2×2分块矩阵结合起来,从而得到了一系列关于矩阵酉不变范数的不等式.
[Abstract]:Matrix theory not only has abundant contents, and is a mathematical tool for a most practical value. In matrix theory, singular value decomposition and singular value inequalities in the high level of statistical calculation and unitarily invariant norm theory plays an important role in. Over the years, many experts and scholars from different angles with different the method of the singular value problems were studied and many marketing, expand the scope of application of the singular values and theoretical system. We use a different method of singular value decomposition and Weyl theorem are proved, and some inference. At the same time unitarily invariant norm as an important numerical characteristics of matrix theory. In the matrix, optimal approximation problem and perturbation theory has an important role. In addition, the block matrix is a special matrix, is an important tool to study the intrinsic properties of the matrix, semi positive definite 2 x 2 points Block matrix and unitary invariant norm together, obtained a series of matrix unitarily invariant norm inequalities. This paper is divided into three chapters, the specific contents are as follows: the first chapter mainly introduces some symbolic concepts involved in this paper, the main normal matrix, unitary matrix, Hadamard product, log- controlled by the weak, weak control, matrix compression, positive semidefinite matrix positive semidefinite square root, generalized inverse matrix, and introduces some known results and theorems. The second chapter gives the singular value decomposition theorem, and then from the spectral decomposition and decomposition of the angle of the other that singular value decomposition method, and obtained the main diagonal element vector the singular value vector and weak controlled relation, and gives other proof of the Weyl theorem and corollaries. Some properties of the third chapter is the unitarily invariant norm and block matrix, the unitarily invariant norm and semi definite 2 * 2 block matrix combination A series of inequalities about the unitary norm of the matrix are obtained.

【学位授予单位】：陕西师范大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O151.21

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