矩阵可逆的若干判别方法.doc

山西师范大学本科毕业论文
矩阵可逆的若干判别方法

姓名
郭晓平
院系
数学与计算机科学学院
专业
数学与应用数学
班级
0701班
学号
0751010139
指导教师
宋蔷薇
答辩日期
成绩
矩阵可逆的若干判别方法
内容摘要
对线性代数和代数学而言,矩阵是一个主要研究对象和重要工具,其中可逆矩阵又是矩阵运算理论的整体不可或缺的一部分。在矩阵理论,可逆矩阵所占的地位是不可替代的,在坐标轴旋转变换公式的矩阵表示、线性变换、线性方程组等理论研究中,它均有重要意义。而且由于在许多有关数学、物理,经济的实际问题中,常常需要通过建立合适的数学模型化为线性代数和代数学等的问题,因此可逆矩阵也是解决实际问题比较常用的工具之一。鉴于可逆矩阵具有重要的理论和实践意义,研究矩阵可逆的判别方法也就相当有必要了。
本文结合所学知识并查阅相关资料,系统地整理并归纳总结了十一种矩阵可逆的判别方法及其证明过程。其中,可逆矩阵判别方法主要包括定义判别法、伴随矩阵判别法、初等变换判别法、线性方程组法、矩阵向量组的秩判别法等。另外,本文还给出了十种特殊矩阵可逆性的相关结论,最后针对这些判别方法选取了典型的例题,以便我们更好的掌握矩阵可逆的判别方法。
【关键词】矩阵逆矩阵初等变换伴随矩阵线性方程组
Some Methods for Judging Invertible Matrix
Abstract
The matrix is a main research subject and an important tool in linear algebra and algebra. The invertible matrix, which plays the role of the invertible number in rational numbers, is an essential part of the matrix theory. The very important status ,which the invertible matrix holds in the matrix theory ,can not be replaced. It has the important meaning for solving linear equations, linear transformation theory problems, rotating coordinate transform formula of matrix representation theory. And In solving practical problems such as mathematics, physics, economic and other fields, it is often need to establish proper mathematical models into linear algebra and algebra issues. Therefore it also is monly used tool, which is widely applied in practical problem. In view of the fact that the invertible matrix has important significance in both theory and practice, the study of judging invertible matrix is quite necessary.
bining with my knowledge, referring to the relevant materials, this paper anizes and summarizes eleven kinds of methods for judging invertible matrix ,which contain definition method, the adjoin matrix method, elementary transformation method, linear equations method and so on ,and the proof process. This paper also gives ten special matrix invertible conclusions. Finally, this paper selects several typical examples aiming at these discriminate