摘要矩阵理论既是经典数学的基础，又是一门很具有实用价值的数学理论。 随着矩阵理论的迅速发展，矩阵的迹在许多领域都有相当多的应用，如逼近论、数值计算、滤波、随机控制和统计估计等等，其许多量的计算都会最终归结到矩阵的迹的运算。关于矩阵的迹的不等式的新结果也是层出不穷，它们有的是经典的不等式的改进、推广，有的则是完全新型的不等式，更有的则是应用的深入和拓广。矩阵的迹已经成为国内外数学学者关注的一个热点研究。82312

本文首先介绍了矩阵的迹的研究现状、定义和其基本性质。然后采用由一般到特殊的方法，介绍了一般的矩阵的迹的性质和相关的推广证明；接着介绍了Hermite矩阵和Neumann矩阵的迹的性质和相关推广证明。最后一部分介绍的是矩阵的迹的应用，从逼近论和统计检验两个方面的应用为例作具体说明。

毕业论文关键词：迹；特征值；Hermite矩阵；Neumann矩阵；不等式

Abstract    Matrix theory is not only the foundation of classical mathematics, but also a mathematical theory with practical value。 With the rapid development of the matrix theory, the trace of a matrix in many areas have quite a number of applications, such as approximation theory, numerical calculation, filtering and stochastic control and statistical estimation and so on, many of its amount of calculation will ultimately attributed to the matrix trace operations。 New results on the trace of matrix inequalities are also emerging。 They have some improvement and generalization of classical inequalities, and others are completely new inequalities。 The trace of matrix has become a hot research topic at home and abroad。

In this paper, we first introduce the research status, definition and basic properties of the matrix trace。 Then by the general to the special method, introduced the general matrix of the trace of the nature and related to the promotion of proof; and then introduced the Hermite matrix and Neumann matrix of the trace of the nature and related to the promotion of proof。 In the last part, the paper introduces the application of the trace of the matrix, and the application of the two aspects of the approximation theory and the statistical test is used as an example to illustrate the application of the theory。

Keywords: Trace;Eigenvalue;Hermite matrix;Neumann matrix;Inequality

目录

第一章 绪论 1

第二章 矩阵迹的一般性质及其推广 3

2。1 迹的定义 3

2。2 迹的性质 3

第三章 Hermite矩阵迹的性质及其推广 6

3。1 定义及相关定理 6

3。2 Hermite矩阵的迹的相关定理 7

3。3 Hermite矩阵迹的推广 8

第四章 Neumann矩阵迹的研究及其推广 11

4。1 两个引理 11

4。2 Neumann矩阵的迹及相关定理 12

4。3 Neumann矩阵迹的相关推论 13

第五章 矩阵幂的迹及其他 15

5。1 矩阵幂的迹 15

5。2 一些其他迹的相关定理 17

第六章 矩阵迹的应用 18

6。1矩阵逼近 18

6。2统计检验中的应用