矩阵加权广义逆与加权极分解研究
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摘要
矩阵的广义逆与极分解在数值分析,矩阵逼近等方面都有很重要的应用,是矩阵理论的重要研究内容。本文主要研究有关矩阵的加权广义逆,加权极分解和矩阵偏序等方面的问题。
普通的矩阵广义逆研究由来已久也趋于成熟。近年来,矩阵的加权广义逆成了矩阵理论研究的热点,许多学者在这个领域做出了一定的成果,我们也得到了一些有意义的结论。我们主要研究了矩阵的加权UDV *分解和加权谱分解以及它们在矩阵方程等方面的应用,探讨了基于加权Moore-Penrose逆的正交投影矩阵的性质及相关扰动界。此外,我们还研究并给出了关于加权广义逆的Lavoie不等式,2×2分块矩阵的加权Moore-Penrose逆的显式表达式等。
矩阵的极分解和广义极分解一直是矩阵分析研究的重要内容,本文中我们对其进行了横向扩展,提出并定义了一种新的极分解形式—矩阵的加权极分解。针对这个新的矩阵分解,我们证明了其唯一性定理,给出了其唯一性条件,讨论了其极因子的最佳逼近性质;同时,我们还探讨了矩阵加权极分解的计算问题,研究了由迭代算法引起的极因子的误差界,极因子在各种范数下的各种形式的扰动界等。在对加权极分解研究的基础上,我们定义并讨论了矩阵的同时加权极分解。
矩阵偏序在数理统计等方面有着重要的应用,是近年来矩阵理论研究的又一热点。本文中我们定义了一种新型矩阵偏序并研究了其基本性质。特别地,与本文提出的矩阵加权极分解和同时加权极分解相结合得到了两个有意义的结论。此外,我们还讨论了几种矩阵加权偏序之间的关系,并结合本文提出的矩阵函数研究了矩阵偏序与其函数偏序之间的关联。
Generalized inverses and polar decomposition of matrices play a significant role in the fields of numerical analysis, matrix approximation and so on. They are all important research topics of matrix theory. In this thesis, we mainly study problems on weighted generalized inverses, weighted polar decomposition, and partial orderings of matrices.
Research on matrix generalized inverses is of long standing and tends to be mature. Recent years, the weighted generalized inverses of matrices become a research hotspot of matrix theory. Many authors made some achievements in this field. We also got some interesting results. We mainly research the weighted UDV* decomposition and weighted spectral decomposition of matrices and its applications in matrix eaquation, and discuss the properties and perturbation bounds of a new type orthogonal projection based on the weighted Moore-Penrose inverse. Furthermore, the Lavoie inequalities for weighted generalized inverses of matrices and an explicit representation of weighted Moore- Penrose inverse of 2×2 block matrices are also studied and presented.
Polar decomposition and generalized polar decomposition of matrices are always the main research subjects of matrix theory. In the present thesis, we generalize them horizontally. A new type of polar decomposition—weighted polar decomposition of matrices is presented and defined. Aim at this new matrix decomposition, we prove its uniqueness theorem and obtain its uniqueness conditions, and also investigate the best approximation property of weighted unitary polar factor. Meanwhile, we also provide methods for computing the weighted polar decomposition, and study error bounds for the approximate generalized positive semidefinite polar factor and perturbation bounds for weighted polar decomposition in various norms. Morevoer, on basis of the weighted polar decomposition, the simultaneous weighted polar decomposition of matrices is also defined and studied.
Matrix partial orderings have many applications in statistics and other fields, and it is a current research focus of matrix theory. In this thesis, we define a new matrix partial ordering and study its basic properties. Especially, combining with the weighted polar decomposition and simultaneous weighted polar decomposition of matrices given in this thesis, we derive two interesting characters of the new matrix partial ordering. Moreover, relations between some matrix weighted partial orderings are investigated, and weighted partial orderings of matrices and orderings of their functions are also compared.
普通的矩阵广义逆研究由来已久也趋于成熟。近年来,矩阵的加权广义逆成了矩阵理论研究的热点,许多学者在这个领域做出了一定的成果,我们也得到了一些有意义的结论。我们主要研究了矩阵的加权UDV *分解和加权谱分解以及它们在矩阵方程等方面的应用,探讨了基于加权Moore-Penrose逆的正交投影矩阵的性质及相关扰动界。此外,我们还研究并给出了关于加权广义逆的Lavoie不等式,2×2分块矩阵的加权Moore-Penrose逆的显式表达式等。
矩阵的极分解和广义极分解一直是矩阵分析研究的重要内容,本文中我们对其进行了横向扩展,提出并定义了一种新的极分解形式—矩阵的加权极分解。针对这个新的矩阵分解,我们证明了其唯一性定理,给出了其唯一性条件,讨论了其极因子的最佳逼近性质;同时,我们还探讨了矩阵加权极分解的计算问题,研究了由迭代算法引起的极因子的误差界,极因子在各种范数下的各种形式的扰动界等。在对加权极分解研究的基础上,我们定义并讨论了矩阵的同时加权极分解。
矩阵偏序在数理统计等方面有着重要的应用,是近年来矩阵理论研究的又一热点。本文中我们定义了一种新型矩阵偏序并研究了其基本性质。特别地,与本文提出的矩阵加权极分解和同时加权极分解相结合得到了两个有意义的结论。此外,我们还讨论了几种矩阵加权偏序之间的关系,并结合本文提出的矩阵函数研究了矩阵偏序与其函数偏序之间的关联。
Generalized inverses and polar decomposition of matrices play a significant role in the fields of numerical analysis, matrix approximation and so on. They are all important research topics of matrix theory. In this thesis, we mainly study problems on weighted generalized inverses, weighted polar decomposition, and partial orderings of matrices.
Research on matrix generalized inverses is of long standing and tends to be mature. Recent years, the weighted generalized inverses of matrices become a research hotspot of matrix theory. Many authors made some achievements in this field. We also got some interesting results. We mainly research the weighted UDV* decomposition and weighted spectral decomposition of matrices and its applications in matrix eaquation, and discuss the properties and perturbation bounds of a new type orthogonal projection based on the weighted Moore-Penrose inverse. Furthermore, the Lavoie inequalities for weighted generalized inverses of matrices and an explicit representation of weighted Moore- Penrose inverse of 2×2 block matrices are also studied and presented.
Polar decomposition and generalized polar decomposition of matrices are always the main research subjects of matrix theory. In the present thesis, we generalize them horizontally. A new type of polar decomposition—weighted polar decomposition of matrices is presented and defined. Aim at this new matrix decomposition, we prove its uniqueness theorem and obtain its uniqueness conditions, and also investigate the best approximation property of weighted unitary polar factor. Meanwhile, we also provide methods for computing the weighted polar decomposition, and study error bounds for the approximate generalized positive semidefinite polar factor and perturbation bounds for weighted polar decomposition in various norms. Morevoer, on basis of the weighted polar decomposition, the simultaneous weighted polar decomposition of matrices is also defined and studied.
Matrix partial orderings have many applications in statistics and other fields, and it is a current research focus of matrix theory. In this thesis, we define a new matrix partial ordering and study its basic properties. Especially, combining with the weighted polar decomposition and simultaneous weighted polar decomposition of matrices given in this thesis, we derive two interesting characters of the new matrix partial ordering. Moreover, relations between some matrix weighted partial orderings are investigated, and weighted partial orderings of matrices and orderings of their functions are also compared.
引文
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陈小山,黎稳. 2005.酉不变范数下极分解的扰动界[J].计算数学, 27:121--128.
陈永林. 2004.用加权MP逆的定义方程推导[ , ]MNA B+的显式[J].南京师大学报(自然科学版), 27:1--5.
董志清,杨虎. 1998. {2}-逆的特性及若干分块问题[J].纯粹数学与应用数学, 14:82--87.
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