本文關(guān)鍵詞：一類(lèi)X型矩陣特征值反問(wèn)題　出處：《大連交通大學(xué)》2015年碩士論文　論文類(lèi)型：學(xué)位論文

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【摘要】：矩陣特征值反問(wèn)題是線性代數(shù)的一個(gè)重要分支,在跨越廣泛的科學(xué)領(lǐng)域中應(yīng)用普遍。自20世紀(jì)50年代第一篇關(guān)于這方面文章發(fā)表以來(lái),越來(lái)越多的此方面研究論文相繼公開(kāi)發(fā)表,獲得了很多深刻而且有益的結(jié)果。當(dāng)今研究的目標(biāo)就是構(gòu)造出了一些實(shí)際科學(xué)應(yīng)用中需要的特征向量和特征值的矩陣。本文研究目標(biāo)定位于構(gòu)造一類(lèi)X型矩陣,從而研究其特征值反問(wèn)題及其廣義特征值反問(wèn)題,利用方程組聯(lián)立求解并遞推得出問(wèn)題解存在并且唯一的條件。全文由以下四章構(gòu)成,內(nèi)容如下:第一章:緒論。首先介紹反問(wèn)題概念、歷史,其次介紹矩陣特征值反問(wèn)題的當(dāng)今研究現(xiàn)狀、難點(diǎn)及未來(lái)應(yīng)用,最后對(duì)本論文所研究的X型矩陣的相關(guān)概念進(jìn)行介紹。第二章:一類(lèi)基本X型矩陣特征值反問(wèn)題。本章首先提出一類(lèi)X型矩陣的特征值反問(wèn)題,并對(duì)矩陣存在的條件進(jìn)行推導(dǎo),得出一類(lèi)X型矩陣特征值反問(wèn)題解存在并且唯一所需要滿足的條件,在此基礎(chǔ)上將X型矩陣右上角的元素剔除,從而得到了我們常見(jiàn)的下三角矩陣,也可稱(chēng)其為退化X型矩陣,按照研究X型矩陣特征值反問(wèn)題的方法,對(duì)此類(lèi)下三角矩陣的特征值反問(wèn)題進(jìn)行研究并得到一類(lèi)退化X型矩陣特征值反問(wèn)題解存在并且唯一所需要滿足的條件并給出解的表達(dá)式。第三章:一類(lèi)特殊退化X型矩陣的特征值反問(wèn)題。本章在第二章中提出的退化X型矩陣的基礎(chǔ)上加以改動(dòng),得到一類(lèi)上三角矩陣,并分別將矩陣元素之間的關(guān)系按照等值關(guān)系、線性關(guān)系分為兩類(lèi),然后對(duì)每一類(lèi)矩陣的特征值反問(wèn)題進(jìn)行研究,分別得到一類(lèi)特殊退化X型矩陣特征值反問(wèn)題解存在并且唯一所需要滿足的條件并給出解的表達(dá)式。最后給出兩個(gè)相應(yīng)的數(shù)值例子分別進(jìn)行了驗(yàn)證。第四章:一類(lèi)特殊退化X型矩陣的廣義特征值反問(wèn)題。本章在前三章的基礎(chǔ)上,將對(duì)矩陣特征值反問(wèn)題的研究擴(kuò)展到對(duì)矩陣廣義特征值反問(wèn)題的研究上,研究了一類(lèi)奇數(shù)階上三角矩陣的廣義特征值反問(wèn)題,得出一類(lèi)特殊退化X型矩陣廣義特征值反問(wèn)題解存在并且唯一的條件并給出解的表達(dá)式。最后給出數(shù)值例子對(duì)算法的有效性進(jìn)行驗(yàn)證。
[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.
【學(xué)位授予單位】：大連交通大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2015
【分類(lèi)號(hào)】：O151.21

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