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  本文關(guān)鍵詞：由譜確定的雙隨機(jī)矩陣和一類矩陣方程問(wèn)題　出處：《中北大學(xué)》2017年碩士論文　論文類型：學(xué)位論文

  更多相關(guān)文章： 由譜確定的雙隨機(jī)矩陣 對(duì)稱M對(duì)稱極小二乘解 對(duì)稱M對(duì)稱最佳逼近解

【摘要】：非負(fù)矩陣逆特征值問(wèn)題一直是數(shù)值代數(shù)中的重點(diǎn)研究對(duì)象,雙隨機(jī)矩陣又是研究矩陣逆特征值問(wèn)題中最常見(jiàn)的矩陣之一,因此研究雙隨機(jī)矩陣自身及其譜的特征是研究其逆特征值問(wèn)題的基礎(chǔ).矩陣方程問(wèn)題來(lái)源于振動(dòng)理論逆問(wèn)題,其主要研究?jī)?nèi)容是求某矩陣方程的不同形式的解及其最佳逼近,并且這一問(wèn)題在機(jī)械系統(tǒng)和土木工程結(jié)構(gòu)中有一定的實(shí)際背景.本文主要研究了一類特殊雙隨機(jī)矩陣的逆特征值問(wèn)題和一類矩陣方程問(wèn)題.全文共分為四章.第一章介紹了本課題的研究意義,以及該課題目前的研究現(xiàn)狀.第二章研究了由譜確定的雙隨機(jī)矩陣的逆特征值問(wèn)題.本章從凸多面體的頂點(diǎn)入手刻畫(huà)了n=3時(shí)由譜確定的雙隨機(jī)矩陣的特征,并針對(duì)一般n階情況時(shí)該矩陣的特征提出了一個(gè)猜想,最后通過(guò)分析置換矩陣的特點(diǎn)及其與兩種n階雙隨機(jī)矩陣之間的關(guān)系,證明了這兩種n階雙隨機(jī)矩陣是由譜確定的.第三章探究了矩陣方程A~TXA=C的對(duì)稱M對(duì)稱極小二乘解及其最佳逼近.本章在對(duì)稱M對(duì)稱矩陣集中,利用典型相關(guān)分解(CCD),獲得了矩陣方程A~TXA=C的對(duì)稱M對(duì)稱極小二乘解;并在給定對(duì)稱矩陣X~*時(shí),應(yīng)用廣義奇異值分解(GSVD)和投影定理,得到了該矩陣方程的對(duì)稱M對(duì)稱最佳逼近解.第四章指出了本文的創(chuàng)新點(diǎn)并對(duì)后續(xù)研究給出了工作展望.
[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.

【學(xué)位授予單位】：中北大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O241.6

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本文編號(hào)：1412163