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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 102093113
指导教师 高菲菲
导师职称 讲师
辩论日期
成绩
容 提 要
矩阵是数学中的一个重要根本概念，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．本文在第一局部介绍了实矩阵的正定性的相关定义以及其等价条件．在第二局部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．本文在第三局部介绍了正定矩阵的相关定理．本文在第四局部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．
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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 formPositive defi