N阶矩阵高次幂的求法及应用

本科毕业论文

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N 阶矩阵m 次方幂的求法及应用

Solution and Application of m-order of n n Martix 作者姓名 指导教师 学科门类 提交论文日期 专业名称

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矩阵是许多实际问题中抽象出来的一个概念，它是高等代数的一个重要组成部分，它几乎贯穿于高等代数的各个章节，在自然学科各分支及经济管理等领域有着广泛的应用.正因为它广泛的应用又是解决众多问题的有力工具，所以，学习并掌握好矩阵的运算以及它们的运算规律和方法是我们学好矩阵知识的一个非常重要的环节.对于矩阵方幂的运算，它是以矩阵的乘法运算为基础；然而，矩阵的幂运算是比较复杂同时也是特别麻烦的，所以寻找简单的运算方法就成了计算矩阵高次幂方面的重要环节，为此很多学者都花了很大的精力去探讨研究，本文将在他们的研究基础上，应用实例通过数学归纳法，乘法结合律的方法，二项式展开式的方法，分块对角矩阵的方法，Jordan标准形法，最小多项式的方法和特殊矩阵法等多种方法来求解方阵的高次幂，进而为n阶矩阵的幂运算来提供一个参考.

关键词:数学归纳法;二项展开式;矩阵的幂;相似矩阵.

Matrix is a concept many practical problems in the abstract, it is an important part of the linear algebra, it is almost throughout the various sections of linear algebra, in the field of natural sciences and economic management of the branch has a wide range of applications. Just because it wide range of applications and is a powerful tool for solving many problems, so learn and master the operation and their method of operation rules and good matrix is a matrix of knowledge we learn a very important part. For matrix power calculations, it is Matrix multiplication is based; however, the matrix exponential operation is more complex but also particularly troublesome, so look for a simple calculation method has become an important part of computing power matrix high regard, for many scholars have spent a lot of research effort to investigate, the paper will be on the basis of their research, application examples by mathematical induction, multiplication associative approach, binomial expansion method, the method block diagonal matrix, standard form method, minimal polynomial a variety of methods and special methods to solve the matrix method phalanx of high-power, and thus the power to order matrix operations to provide a reference.

Keywords：Mathematical induction; power matrix;; binomial expansion similar matrix .

目 录

摘 要.............................................................. I Abstract........................................................... I I 目 录............................................................ I II 引 言............................................................... 1 1 准备知识. (1)

2.1 利用数学归纳法求解n 阶矩阵的高次幂 .......................... 2 2.2利用二项式展开法求矩阵的高次幂 .............................. 4 2.3 利用Jordan 标准形求矩阵的高次幂 ............................ 5 2.4 利用分块对角矩阵求矩阵的高次幂............................. 8 2.5 利用乘法结合律求方阵的高次幂............................... 10 2.6 利用最小多项式解矩阵的高次幂.............................. 11 2.7 利用特殊矩阵法求解矩阵的高次幂. (13)

2.7.1 对合矩阵............................................ 13 2.7.2 幂等矩阵............................................. 14 2. 8 利用图论算法求矩阵的高次幂. (15)

2.8.1 邻接矩阵............................................ 15 2.8.2 n n A AAA

AA 的元素的意义 (15)

2.9利用特征多项式求解矩阵的高次幂 ............................. 16 3 矩阵的幂在人口流动的中的应用..................................... 17 总 结.............................................................. 20 参考文献........................................................... 21 致 谢.. (22)