Research On Weighted Generalized Inverses And Weighted Polar Decomposition Of Matrices

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.