关于特殊Toeplitz矩阵和拟Toeplitz矩阵逆的研究

发布时间：2018-04-01 15:15

本文选题：结构矩阵　切入点：行列式　出处：《山东师范大学》2017年硕士论文

【摘要】：本文研究了七种结构矩阵(含有著名数的斜对称Toeplitz矩阵,列上加下Toeplitz矩阵与列上加下Hankel矩阵,列上减下Toeplitz矩阵与列上减下Hankel矩阵,列上加r下Toeplitz矩阵与列上加r下Hankel矩阵)的行列式、逆矩阵及稳定性分析等问题,分为以下六章:第一章分为三部分,第一部分介绍了循环矩阵、Toeplitz矩阵等几种结构矩阵的应用背景及国内外研究现状,第二部分介绍了七种特殊结构矩阵的定义及相关引理,第三部分就本文的主要工作进行了简单的介绍.第二章主要考虑了包含有著名数的Toeplitz矩阵的行列式和逆,共分为三部分.第一部分通过构造合适的变换矩阵得到Fibonacci斜对称Toeplitz矩阵的行列式和逆矩阵.第二部分,构造相似的变换矩阵,得到Lucas斜对称Toeplitz矩阵的行列式和逆矩阵.第三部分给出例子来验证我们得到的结论.第三章、第四章与第五章分别以列上加下Toeplitz矩阵(TCUPL)和列上加下Hankel矩阵(HCUPL)、列上减下Toeplitz矩阵(T_(CUML))和列上减下Hankel矩阵(H_(CUML))与列上加r下Toeplitz矩阵(T_(CUPrL))和列上加r下Hankel矩阵(H_(CUPrL))这六种结构矩阵为对象,研究了这六种结构矩阵的逆矩阵的分解式,之后给出了算法,并对其稳定性进行了分析.并在第五章给出例子来验证我们得到的定理.第六章对本文的主要工作进行了概括与总结,并对将来的工作进行了展望.
[Abstract]:In this paper, we study seven kinds of structural matrices (skew symmetric Toeplitz matrices with famous numbers, upper and lower Toeplitz matrices, column upper and lower Hankel matrices, column upper and lower Toeplitz matrices and column upper and lower Hankel matrices). The determinant, inverse matrix and stability analysis of Toeplitz matrix and Hankel matrix are divided into the following six chapters: the first chapter is divided into three parts. In the first part, the application background of several kinds of structure matrices, such as cyclic matrix and Toeplitz matrix, and the research status at home and abroad are introduced. In the second part, the definitions of seven kinds of special structural matrices and their relevant Lemma are introduced. In the third part, the main work of this paper is briefly introduced. In chapter 2, determinant and inverse of Toeplitz matrix with famous number are considered. In the first part, the determinant and inverse matrix of Fibonacci skew symmetric Toeplitz matrix are obtained by constructing suitable transformation matrix. In the second part, the similar transformation matrix is constructed. The determinant and inverse matrix of Lucas skew symmetric Toeplitz matrix are obtained. In chapters 4 and 5, respectively, the six structural matrices of the column with the addition of the Toeplitz matrix and the column with the addition of the Hankel matrix, the upper and the lower Toeplitz matrices are taken as objects, respectively, and the Hankel matrices of the upper column and the lower column subtract the Hankel matrix, and the Toeplitz matrix with the addition of r is added to the column, and the Toeplitz matrix with the addition of r to the column, and the Hankel matrix under the column is added to the column as objects. In this paper, the factorization of the inverse matrix of these six kinds of structural matrices is studied, and the algorithm is given. In chapter 5, an example is given to verify the obtained theorem. In chapter 6, the main work of this paper is summarized and the future work is prospected.
【学位授予单位】：山东师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O151.21

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