【作者】 朱辉华；

【导师】 刘建州；

【作者基本信息】 湘潭大学 ， 应用数学， 2007， 硕士

【摘要】 特殊矩阵在矩阵分析和矩阵计算中具有重要的意义。它在计算数学、经济学、生物学、应用数学等领域都有着广泛的应用，逆M-矩阵是一类重要的特殊矩阵。本文利用逆M-矩阵的性质，探索了几类特殊形状矩阵的结构，得到了它们是逆M-矩阵的充分或充要条件。第一章主要介绍了特殊形状和特殊类型矩阵近期的研究成果，特别简述了逆M-矩阵的研究成果。第二章将矩阵进行特殊分块，结合schur补矩阵的性质，得到了非负矩阵是逆M-矩阵的充要条件；进一步结合周期三对角矩阵的性质和三对角逆M-矩阵充要条件，得到了周期三对角逆M-矩阵的充要条件。在此基础上，我们讨论了更为广泛的一类矩阵-加元周期三对角矩阵，相应地获得了加元周期三对角矩阵是逆M-矩阵的充要条件，并且给出了相应的数值实例。第三章在第二章的基础上，我们定义了加多元周期三对角矩阵，讨论了其性质，并得到了加多元周期三对角逆M-矩阵的一个充分条件；给出了相应的数值实例。第四章利用矩阵分块方法，结合五对角矩阵的结构性质，获得了五对角逆M-矩阵的一个充分条件；进一步证明了这类矩阵在Hadamard积下的封闭性。更多还原

【Abstract】 Special matrices play important roles in matrix analysis and matrix computation and have wide applications in computational mathematics, economics, biology, applied mathematics and etc. Inverse M—matrices is one of the most important special matrices. In this paper, by using the properties of inverse M—matrices, and investigating the structure of some kinds of special shape matrices, we get the sufficient and necessary or sufficient conditions of them.In chapter one, we mainly introduce the studied production of the specific shape and type matrix in the near future, and state the research production of inverse M—matrices.In chapter two, partitioning specially and using the properties of schur complements matrices, we get the necessary and sufficient conditions of inverse M—ma trices; the properties of tridiagonal period inverse M-matrices and the sufficient and necessary of the tri-diagonal matrices are associated further, we get the necessary and sufficient condition of the tridiagonal period inverse M—matrices. And we discuss a new class of adding element tri-diagonal period inverse M—matrices that has analogous properties, the necessary and sufficient condition of adding element tri-diagonal period inverse M—Matrices is given, moreover; we give relevant numerical example.In chapter three, based on the chapter one, we defined a new class of adding many elements tri-diagonal period matrices, on the blocked matrices above, integrating the properties of adding element tridiagonal period inverse M—matrices we get a sufficient condition of adding many elements tri-diagonal period inverse M—Matrices, moreover the numerical example is given.In chapter four, using partitioning of block matrix, and associating the properties of five-diagonal inverse M—matrices matrices, we get a sufficient condition of five-diagonal inverse M—matrices, and we prove that the class of five-diagonal inverse M—matrices matrices is closed under Hadamard product.更多还原