【作者】 韩俊林；

【导师】 刘建州；

【作者基本信息】 湘潭大学 ， 应用数学， 2002， 硕士

【摘要】 各类对角占优矩阵及分块对角占优矩阵在矩阵理论及计算数学、经济学、统计学等实际应用中具有重要地位，国内外不少学者都作了许多重要工作。本文将对它们的判别、性质及应用作进一步的讨论。 第一章，我们先简介了几类对角占优阵的定义及它们之间的关系，并介绍了一些基本迭代法(指Jacobi,Seidel及SOR迭代法等)的收敛性与这些矩阵类的关系。接着，介绍了几类块对角占优矩阵的定义及它们之间的关系。 第二章，简介了判别广义对角占优阵的一些方法，指出广义次对角占优矩阵与广义对角占优矩阵的关系，并用较简捷的方法证明了广义对角占优阵与广义次对角占优阵的一个等价条件。 第三章，研究了几类对角占优矩阵及M矩阵的Kronecker和的性质，并对矩阵方程AX+XB=C在特殊情形下的解法进行了探讨。 第四章，将块对角占优阵与Khatri-Rao积相结合，获得了一些重要结果。更多还原

【Abstract】 Several kinds of diagonally dominant matrices and block diagonally dominant matrices play an important role in numerical mathematics, economics and statistics, many scholars have gotten important conclusions on them. In this paper, further research are carried out on their identification, properties and applications.In chapter one, firstly, we introduced several kinds of diagonally dominant matrices’ definition and their connection. Secondly, the relationship between those kinds of matrices and the convergence of some basic iteration methods (such as Jacobi, Seidel and SOR iteration methods) are also mentioned. Thirdly, several kinds of block diagonally dominant matrices’ definition and their connection are introduced.In chapter two, we recommend some methods for how to identify generalized diagonally dominant matrices, point out the relationship between generalized diagonally dominant matrices and generalized subdiagonally dominant matrices, and besides, we give a succinct proof to an equivalence condition of the generalized diagonally dominant matrices.In chapter three, we discussed some properties of Kronecker sum of some matrices, such as M-matrices, H-matrices, diagonally dominant matrices and a - diagonally dominant matrices, and study the solution of the matrix equation AX+XB=C.In chapter four, block diagonally dominant matrices and block diagonally dominant matrices are combined, some important results are got.更多还原