矩阵指数函数的性质与计算

矩阵指数函数的性质与计算PROPERTIES AND CALCULATION OF MATRIX EXPONENTIAL FUNCTION

指导教师：

申请学位级别：学士

论文提交日期：2014年6月 8日

摘要

矩阵函数是矩阵理论的重要组成部分，而矩阵函数中的一个最重要的函数就是矩阵指数函数，它广泛地应用于自控理论和微分方程。本文深入浅出地介绍了矩阵指数函数，并进一步探讨如何借助矩阵指数函数分析相关问题。文章以齐次线性微分方程组求解基解矩阵为出发点引出矩阵指数函数的概念，证明求解矩阵指数函数就是求解齐次线性微分方程组的基解矩阵，然后得到矩阵指数函数的一些基本性质。本文的重点是讨论矩阵指数函数的五种计算方法。其中，前三种方法广泛适用于各种矩阵，虽然计算过程复杂程度不同，但都需要计算矩阵特征值，如遇高阶矩阵或复特征值，则特征值的计算会变得异常麻烦。后两种方法较特殊，虽然缺乏普适性，只能计算特殊矩阵的指数函数，但却避过了特征值计算，简化了运算过程。最后，本文具体阐述矩阵指数函数在微分方程求解中的应用。

关键词：矩阵指数函数；Jordon 标准形；微分方程组

ABSTRACT

Matrix function is an important part of the matrix theory. And among the matrix function, there is a special and important function that is matrix exponential function. It has been widely used in automatic control theory and differential equations. This paper introduces profound theories on matrix exponential function in simple language, furthermore, it explores how to use matrix exponential function analysis related issues. Through the basic solution matrix of homogeneous linear differential equations, this paper draws out the concept of matrix exponential function. In this part, the author proves that solving matrix exponential function is to solve the basic solution matrix of the homogeneous linear differential equations. Then, some basic properties of matrix exponential function can be derived. The focus of this paper is on the discussion of five kinds of calculation on matrix exponential function. The first three methods can be applied to general cases. Although each method is different, in complexity, all of them need to compute the matrix eigenvalues. The calculation on high-order matrix or complex eigenvalues will be in trouble frequently. The latter two methods is more special for they can only calculate special matrix exponential function. These methods simplify the operation process instead of calculating eigenvalues, but their shortcomings are obvious. At the final part of this paper, the article expounds the application of

matrix exponential function in different equations when solving the function in reality.

Key words: Matrix exponential function; Jordon normal form; Differential equations

目录

1 前言 (1)

1.1 矩阵（Matrix）的发展与历史 (1)

1.2 本文的主要容 (2)

2 预备知识 (3)

3 矩阵指数函数的性质 (7)

3.1 矩阵指数 (7)

3.1.1 关于级数

! k k

k A t k

∞

=

∑的收敛性 (7)

3.1.2 矩阵指数A e的性质 (8)

3.1.3 常系数线性微分方程基解矩阵 (10)

3.2 矩阵指数函数的性质 (130)

3.2.1 矩阵函数 (130)

3.2.2 矩阵指数函数的性质 (141)

4 矩阵指数函数的计算方法 (207)

4.1 矩阵指数函数的一般计算方法 (207)

4.1.1 Hamilton‐Cayley求解法 (217)

4.1.2 微分方程系数求解法 (251)

4.1.3 Jordon块求解法 (283)

4.2 矩阵指数函数的特殊计算方法 (316)

4.2.1 矩阵指数函数展开法 (327)

4.2.2 Laplace变换法 (27)

4.3 矩阵指数函数方法比较 (28)