新的矩阵对角占优形式与M—矩阵和H矩阵的研究

【作者】 房秀芬；

【导师】 黄廷祝；

【作者基本信息】 电子科技大学 ， 应用数学， 2000， 硕士

【摘要】 M—矩阵和H矩阵是计算数学的重要分支——数值代数研究的重要矩阵类。对于M—矩阵和H矩阵的研究在计算数学和其他许多应用领域中起着非常关键的作用。 本文着重对于这两个重要矩阵类的概念、判定以及等价表征作进一步研究。提出了一些新的广义矩阵对角占优概念，籍之研究M—矩阵和H矩阵，获得了关于M—矩阵和H矩阵的若干实用简捷判据，所获结果不要求矩阵预先对角占优，且非对角占优行可达n-1行，适用于广泛的矩阵类。 论文分为4章： 第一章，给出了本论文所用到的关于矩阵理论中的一些基本概念； 第二章，提出了不可约共轭对角占优矩阵和具非零元素链共轭对角占优矩阵的定义，研究了它们的谱性质，也得到了H矩阵的两个子类； 第三章，提出了新的矩阵对角占优概念——局部双α对角占优矩阵，通过对局部双α对角占优矩阵的研究，获得了M—矩阵和H矩阵的新的简单实用判据，该判据只需矩阵元素的简单代数关系式，而且不要求矩阵预先对角占优，且非对角占优行可达n-1行，适用于广泛的矩阵类； 第四章，提出了新的矩阵对角占优概念——加权连对角占优矩阵，通过对加权连对角占优矩阵的研究，得到了M—矩阵的新的简单充要条件。更多还原

【Abstract】 M-matrices and H matrices are important matrix species of the research for numeric algebra -the important branch of computational mathematics. The research for M-matrices and H matrices has too much pivotal action in computational mathematics and the other lots of applied scopes. In this paper, we put emphasis on introducing the concept, determinant and equivalent typicality of the two important matrix species. We provide some new concept of generalized diagonally dominant matrices to research M- matrices and H matrices. And we obtain some practical brief determinant for M-matrices and H matrices. The results don’t request that the matrices must be predetermined diagonally dominant, and non-diagonally dominant rows can get n-I. It’s adapted to far-ranging matrix species. In this paper, four chapters are included: In chapter 1, some basic concepts about matrix theory using in this dissertation are introduced; In chapter 2, the definitions of irreducibility conjugacy diagonally dominant matrix and conjugacy diagonally dominant matrix with nonzero element chain is provided, and we obtain two subclasses of H matrices; In chapter 3, a new concept of diagonally dominant matrix art double a diagonally dominant matrix is provided, and we obtain new practical brief determinant for M-matrices and H matrices through the research for part double a diagonally dominant matrix. This determinant only need brief algebra connecting expression, and don’t request that the matrices must bepredetermined diagonally dominant, and non-diagonally dominant rows canget n- l. It’s adapted to ttrr-ranging matrix species.ln chapter 4, a new concept of diagonally dominant matrix-weightedconnective diagonally dominant matrix is provided, and we obtain new briefsufficient and need conditions of M-matrices through the research fOr weightedconnective diagonally dominant matrix.更多还原