关于反对称反循环矩阵相关性质的研究
Study on Skew-Symmetric and Skew-Circulant Matrices
【中文摘要】 循环矩阵是矩阵理论的一个重要组成部分,且日益成为应用数学领域中一个非常活跃和重要的研究方向。反对称反循环矩阵又是循环矩阵的一个重要组成部分。它具有许多特殊的性质和结构,因此很有必要对其进行研究,并探讨其特殊性质和特殊结构。例如:各种多项式表示形式、对角化、谱分解、非奇异性、特征值、特征多项式、极小多项式、逆阵、群逆及Moore-Penrose逆的各种快速算法等。本文主要研究内容如下:首先给出了反对称反循环矩阵的定义并利用Vandermonde矩阵讨论了反对称反循环矩阵的准对角化问题;并由所得到的结果,获得了反对称反循环矩阵的一些相关性质,进而给出了一种简便的反对称反循环矩阵求逆的算法。其次,将反对称反循环矩阵进行了推广,得到了几种分块反对称反循环矩阵,并对其中的两种特殊分块的反对称反循环矩阵的性质进行了讨论。最后,在分块反对称反循环矩阵性质的基础上,给出了其特征值和特征多项式以及相似对角阵。本文共分三个部分:一:给出相关的预备知识,主要是循环矩阵研究的国内外现状和进展、文中用到的循环矩阵的基本概念、性质以及在矩阵理论和矩阵计算中经常用到基本运算工具。二:给出了一个新的矩阵类型-反对称反.循环矩阵的概念,并给出了该矩阵的一系列性质,以及利用Vandermonde矩阵将反对称反循环矩阵对角化,并给出反对称反循环矩阵求逆的方法以及逆矩阵的性质。三:给出了两种分块不同的块反对称反循环矩阵的概念,并对他们的性质进行了研究,给出了相关的结论。 

【英文摘要】 The study of circulant matrices, an important component of the matrix theory, has become one of the most important and active research fields in applied mathematics. Skew-symmetric and skew-circulant matrices studies are basilic part of circulant matrix. Due to the special features of skew symmetric and skew-circulant matrices, it’s necessary for us to generalize their unique structures and characteristics, such as: all kinds of polynomial representations, diagonalization, special decomposition, nonsingularity eigenvalues, characteristic polynomial and fast algorithms for computing minimal polunomial, inverse, self-reflective g -inverse, group inverse and Moore-Penrose inverse, and so on. The main contents of this paper are as follows:Firstly, skew-symmetric and skew-circulant matrix was defined, its diagonalization by using Vandermonde matrix was discussed, through which some results were given. With these results, related features of skew-symmetric and skew-circulant matrix were dedu.ced. And in this way, a simple algorithm of reversing of Skew-symmetric and skew-circulant matrices were given. Secondly, with the development of the skew symmetric and skew circulant matrix, several block skew-symmetric and skew-circulant matrix were defined, two of which were discussed in terms of features and characteristics. Finally, based on these characteristics , the eigenvalues and eigenvalues polynomials and its diagonal matrix were given.This article is divided into three parts:I: It gives the relevant background knowledge, mainly about the study of circulant matrices at home and abroad, the basic concepts, characteristics of circulant matrices, and the basic computing instruments which have been frequently used in matrix theory and matrix calculations.II: It gives a new matrix type-- the skew-symmetric and skew-circulant matrix and a series of the characteristics of this matrix, and comes up with its diagonalization using the Vandermonde matrix. Then it gives the reverse method of the skew-symmetric and skew-circulant matrix, and the characteristics of the inverse matrix.III: In this paper, the concepts of two different sub-block of block skew-symmetric and skew-circulant matrix was given, and their characters were discussed, relevant conclusions were presented. 

【中文关键词】 反对称反循环矩阵; 对角化; Vandermonde矩阵; 特征值; 逆矩阵
【英文关键词】 skew-symmetric and skew-circulant matrices; diagonalization ; Vandermonde matrix ; eigenvalue ; inverse matrix
【毕业论文目录】
摘要 4-5
ABSTRACT 5
目录 6-7
1 绪论及预备知识 7-17
    1.1 循环矩阵的发展和研究现状 7-13
        1.1.1 几类循环矩阵的概念 8-10
        1.1.2 基本循环矩阵类型 10-12
        1.1.3 基本循环矩阵的性质 12-13
    1.2 循环矩阵的基本运算 13-16
        1.2.1 循环矩阵的直和 13-14
        1.2.2 矩阵的Kronecker 积 14-16
        1.2.3 Vandermonde 矩阵 16
    本章小结 16-17
2 反对称反循环矩阵 17-40
    2.1 反对称反循环矩阵的概念 17-23
        2.1.1 反对称反循环矩阵多项式 18-23
    2.2 反对称反循环矩阵的性质 23-30
    2.3 反对称反循环矩阵的对角化问题 30-33
    2.4 反对称反循环矩阵的逆矩阵 33-37
        2.4.1 反对称反循环逆矩阵的性质 33-35
        2.4.2 反对称反循环逆矩阵的求法 35-37
    2.5 反对称反循环矩阵的实例计算 37-40
3 分块反对称反循环矩阵及其性质 40-53
    3.1 一般分块反对称反循环矩阵 40-43
        3.1.1 一般分块反对称反循环矩阵的概念 40-41
        3.1.2 分块反对称反循环矩阵的性质 41-43
    3.2 块为对称阵的分块反对称反循环矩阵 43-48
        3.2.1 块为对称阵的分块反对称反循环矩阵性质 43-44
        3.2.2 块为对称阵的分块反对称反循环矩阵的特征值 44-48
    3.3 二重反对称反循环矩阵 48-53
        3.3.1 二重反对称反循环矩阵的概念 48-49
        3.3.2 二重反对称反循环矩阵的性质 49-53
结论 53-54
参考文献 54-58
致谢 58-59
作者硕士期间成果 59-60