学院2016届
本科毕业论文(设计)
矩阵的对角化及其应用
学生姓名： 学 号：
专 业： 数学与应用数学 指导老师： 答辩时间： 2016.5.22 装订时间： 2016.5.25
A Graduation Thesis (Project)
Submitted to School of Science, Hubei University for Nationalities
In Partial Fulfillment of the Requiring for BS Degree
In the Year of 2016
Diagonalization of the Matrix and its Applications
Student Name Student No.:
Specialty: Mathematics and Applied Mathematics Supervisor:
Date of Thesis Defense:2016.5.22 Date of Bookbinding: 2016.5.25
摘 要
矩阵在大学数学中是一个重要工具，在很多方面应用矩阵能简化描述性语言，而且也更容易理解，比如说线性方程组、二次方程等. 矩阵相似是一个等价关系，利用相似可以把矩阵进行分类，其中与对角矩阵相似的一类矩阵尤为重要，这类矩阵有很好的性质，方便我们解决其它的问题. 本文从矩阵的对角化的诸多充要条件及充分条件着手，探讨数域上任意一个n阶矩阵的对角化问题，给出判定方法，研究判定方法间的相互关系，以及某些特殊矩阵的对角化，还给出如幂等矩阵、对合矩阵、幂幺矩阵对角化的应用.
关键词：对角矩阵，实对称矩阵，幂等矩阵，对合矩阵，特征值，特征向量，最小多项式
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Abstract
The matrix is an important tool in college mathematics, and can simplify the description language based on the application of matrix in many ways. So it is easier to understand in many fields, for example, linear equations, quadratic equations. In many characteristics, the matrix similarity is an very important aspect. We know that the matrix similarity is an equivalence relation by which we can classify matrix, the diagonal matrix is very important. This kind of matrix has good properties, and it is convenient for us to solve other problems, such as the application of similar matrix in linear space. In this paper, we first discuss many necessary and sufficient conditions of diagonalization of matrix and then give some applications of special matrix diagonalization.
Key words: diagonal matrix，real symmetric matrix，idempotent matrix，involutory
matrix，the eigenvaule，the feature vector，minimal polynomial
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目 录
摘要??????????????????????????????????I
Abstract????????????????????????????????II
绪言??????????????????????????????????1
课题背景????????????????????????????????1
目的和意义?????????????????????????????? 1
国内外概况?????????????????????????????? 1
预备知识????????????????????????????????2
相关概念????????????????????????????????2
矩阵的对角化??????????????????????????????4
特殊矩阵的对角化??????????????????????????? 14
矩阵对角化的应用??????????????????????????? 22
总结????????????????????????????????? 24
致谢????????????????????????????????? 25
参考文献??????????????????????????????? 26
独创声明??????????????????????????????? 28
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