正定矩阵的性质和判定方法及应用

内蒙古财经大学本科毕业论文正定矩阵的性质及应用

作者郝芸芸

系别统计与数学学院

专业信息与计算科学

年级10级

学号*********

指导教师高菲菲

导师职称讲师

答辩日期

成绩

内容提要

矩阵是数学中的一个重要基本概念，也是一个主要研究对象，同时矩阵论又是研究线性代数的一个有力工具．而矩阵的正定性是矩阵论中的一个重要概念．正定矩阵是一种特殊的矩阵，其等价定理在解题过程中可以灵活使用．且正定矩阵具有一般矩阵不具有的特殊性质，尤其是这些性质广泛地应用于各个领域．本文在第一部分介绍了实矩阵的正定性的相关定义以及其等价条件．在第二部分列举了正定矩阵的一系列性质，主要介绍了正定矩阵的关联矩阵的正定性．本文在第三部分介绍了正定矩阵的相关定理．本文在第四部分介绍了矩阵正定性的判定方法：定义法、主子式法、特征值法、与单位矩阵合同法．且简单地举了一些实例来阐述实矩阵正定性的判定．最后本文分别从不等式的证明和多元函数的极值两个方面介绍了正定矩阵的实际应用．

关键词：二次型正定矩阵判定方法应用

Abstract

Matrix is an important basic concepts in mathematics, but also a main research object, at the same time matrix theory is a powerful tool for the study of linear algebra. At the same time, the positive definiteness of matrix is an important concept in the matrix theory. The positive definite matrix is a special matrix, the equivalence theorem in the problem solving process can be used flexibly. And the positive definite matrix with special properties of general matrix does not have these properties, especially widely used in various fields. In the first part of this thesis introduces the related definition of positive definite real matrix and its equivalent conditions. In the second part are held a series of properties of positive definite matrix, mainly introduced the positive definiteness correlation matrix is positive definite matrix. This paper introduces the related theorem of positive definite matrix in the third part. This paper introduces the method to judge the positive definiteness matrix in fourth parts: the definition, the master method, the eigenvalue method. Determination and simply cited a number of examples of real positive definite matrices. Two aspects of extreme finally this paper from the proof of inequality and multiple function describes the practical application of positive definite matrices.

Key words:Quadratic form Positive definite matrix Determination method Application

目录

引言...................................................... 错误!未定义书签。

一、正定矩阵的定义........................................ 错误!未定义书签。

二、正定矩阵的性质........................................ 错误!未定义书签。

三、正定矩阵的有关定理.................................... 错误!未定义书签。

四、正定矩阵的判定方法.................................... 错误!未定义书签。（一）定义法............................................ 错误!未定义书签。（二）主子式法.......................................... 错误!未定义书签。（三）特征值法.......................................... 错误!未定义书签。（四）与单位矩阵E合同法................................ 错误!未定义书签。

五、正定矩阵的应用........................................ 错误!未定义书签。（一）正定矩阵在不等式中的应用.......................... 错误!未定义书签。（二）正定矩阵在多元函数极值问题中的应用................ 错误!未定义书签。总结...................................................... 错误!未定义书签。参考文献.................................................. 错误!未定义书签。后记...................................................... 错误!未定义书签。