摘要矩阵的Kronecker积是任意的两个矩阵之间的乘积运算。 这种乘积没有一般的两个矩阵乘积可相乘的条件为前面矩阵的列数必须等于后面矩阵的行数，否则就无法进行乘积运算的限制，为我们的研究提供了方便。 而且矩阵的Kronecker积有着重要的应用价值，在矩阵线性方程和矩阵微分运算的讨论中，应用Kronecker积常会使运算方便简洁。同时矩阵的Kronecker积在诸如信号处理与系统控制等工程领域中也是一种基本的数学工具。 因此对矩阵的Kronecker积的研究具有重要的理论价值和实际应用价值，并且有着广泛的应用前景。86707

本次论文主要研究矩阵的Kronecker积，总结了矩阵的Kronecker积的基本性质，着重研究其不变性以及某些特殊矩阵Kronecker积的性质。 并且给出了矩阵的Kronecker积的应用,如求解一般线性矩阵方程、矩阵微分方程及矩阵函数积的导数。

毕业论文关键词：Kronecker积；不变性；线性矩阵方程；矩阵微分方程；导数

Abstract Kronecker product of a matrix is a product of arbitrary two matrices。 This product does not have a general of product of two matrices multiplication conditions for the number of columns in front of the matrix must be equal to the number of rows of the matrix behind, otherwise will not be able to carry out multiplication, and provides convenience for our study。 And the matrix Kronecker product has important application value, in the matrix linear equation and matrix differential operation of the discussion, the application of Kronecker product often makes the operation convenient and concise。 At the same time, the Kronecker product of the matrix is also a basic mathematical tool in the engineering fields such as signal processing and system control。 So it has important theoretical value and practical application value to study the Kronecker product of the matrix, and has a broad application prospect。

In this paper, we mainly study the Kronecker product of the matrix, and summarize the basic properties of the matrix Kronecker product, and focus on the properties of the Kronecker product of some special matrices。 And the application of the Kronecker product of the matrix is given, such as the solution of the general linear matrix equation, the matrix differential equation and the derivative of the matrix function product。

Keywords: Kronecker Product; Invariance; Linear Matrix Equation; Matrix Differential Equation; Derivative

目  录

第一章 绪论 1

1。1 研究背景及意义 1

1。2 研究现状 1

1。3 本文主要内容 1

第二章 矩阵的Kronecker积 3

2。1 基本概念 3

2。2 本章小结 3

第三章 矩阵的Kronecker积的性质 4

3。1 矩阵的Kronecker积的性质 4

3。1。1 基本性质 4

3。2 本章小结 7

第四章 特殊矩阵的kronecker积的性质 8

4。1 常见特殊矩阵 8

4。2 亚正定矩阵 10

4。3 k-Hermitian矩阵 11

4。4 置换矩阵 15

4。5 本章小结 16

第五章 矩阵的Kronecker积的应用 17

5。1线性矩阵方程的求解