由谱确定的双随机矩阵和一类矩阵方程问题

发布时间：2018-01-12 01:37

本文关键词：由谱确定的双随机矩阵和一类矩阵方程问题　出处：《中北大学》2017年硕士论文　论文类型：学位论文

【摘要】：非负矩阵逆特征值问题一直是数值代数中的重点研究对象,双随机矩阵又是研究矩阵逆特征值问题中最常见的矩阵之一,因此研究双随机矩阵自身及其谱的特征是研究其逆特征值问题的基础.矩阵方程问题来源于振动理论逆问题,其主要研究内容是求某矩阵方程的不同形式的解及其最佳逼近,并且这一问题在机械系统和土木工程结构中有一定的实际背景.本文主要研究了一类特殊双随机矩阵的逆特征值问题和一类矩阵方程问题.全文共分为四章.第一章介绍了本课题的研究意义,以及该课题目前的研究现状.第二章研究了由谱确定的双随机矩阵的逆特征值问题.本章从凸多面体的顶点入手刻画了n=3时由谱确定的双随机矩阵的特征,并针对一般n阶情况时该矩阵的特征提出了一个猜想,最后通过分析置换矩阵的特点及其与两种n阶双随机矩阵之间的关系,证明了这两种n阶双随机矩阵是由谱确定的.第三章探究了矩阵方程A~TXA=C的对称M对称极小二乘解及其最佳逼近.本章在对称M对称矩阵集中,利用典型相关分解(CCD),获得了矩阵方程A~TXA=C的对称M对称极小二乘解;并在给定对称矩阵X~*时,应用广义奇异值分解(GSVD)和投影定理,得到了该矩阵方程的对称M对称最佳逼近解.第四章指出了本文的创新点并对后续研究给出了工作展望.
[Abstract]:The inverse eigenvalue problem for nonnegative matrix has been a key research object in numerical algebra, doubly stochastic matrix is one of the most common matrix inverse eigenvalue problem of matrix, so the study of doubly stochastic matrix and its spectrum characteristics is based on the inverse eigenvalue problem of matrix equation. The problem comes from the theory of vibration inverse the problem, the main research content is to find a matrix equation of the different forms of solutions and the optimal approximation, and has certain practical background of this problem in the mechanical system and civil engineering structures. This paper mainly studies the characteristics of the inverse of a class of special two random matrix problems and a class of matrix equation problem. It consists of for the four chapter. The first chapter introduces the research significance of this topic, and the research status at present. The second chapter studies the value problem of inverse eigenvalue spectrum determined by double random matrix. This chapter from the convex polyhedron The characteristics of the vertex doubly stochastic matrix is determined by the spectral characterization of n=3, and puts forward a conjecture for the feature matrix of order n general situation, finally through analyzing the characteristics of permutation matrix and the relationship between the two order n doubly stochastic matrix, it is proved that the two kinds of order n doubly stochastic matrix by the spectrum determined. The third chapter explores the symmetric M symmetric matrix equation A~TXA=C the least-squares solutions and the optimal approximation. This chapter in the symmetric M symmetric matrix, using the canonical correlation decomposition (CCD), the symmetric M symmetric matrix equation of A~TXA=C least squares solution; and in a given symmetric matrix X~*. The application of the generalized singular value decomposition (GSVD) and the projection theorem, the symmetric matrix equation M symmetric optimal approximation solution. The fourth chapter points out the innovation of this paper and prospects for future research were given.

【学位授予单位】：中北大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.6

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