Research On The Inequalities Of Matrix Partial Order And Special Matrix

Matrix partial ordering is a significant branch of matrix theory,it not only has a lot of research contents,but also is an important tool for studying many branches of mathematics and other natural science.Matrix inequality that plays a very important role in practical applications and calculations is another important research direction of matrix theory.Based on the partial orders and inequalities of matrix studied by experts and scholars,this paper mainly study Lowner partial order,star partial order,minus partial order,and partitioned positive semi-definite matrix and normal matrix,and got some important conclusions.The main contents of this paper can be summarized as follows:The first chapter mainly introduces some basic symbols,concepts,and some theorems used in this paper.For example the definitions of normal matrix,Schur complementary matrix,generalized inverses of matrix and the theorems of Schur complementary theorem,the inverse theorem of partitioned matrix.In chapter two,based on the principle that the main sub-matrix of the inverse matrix of positive definite matrices is always not lower than the inverse of the main sub-matrix corresponding to the original square matrix in the sense of Lowner partial order,firstly,the Lowner partial order of main sub-matrix is researched by the matrix Schur decomposition,and some inequalities about the Lowner partial order of the main sub-matrix are also obtained;Secondly,the star partial order definition and equivalent conditions are studied,some basic properties and heredity of the star partial order are obtained,and the properties of these partial orders for the minus partial order and Lowner partial order are also established;Finally,using the decomposition of the matrix star order partial,minus order partial and matrix singular value decomposition,we study the relationship between Lowner partial order,star partial order and minus partial order,and gained the connection between the three.In this chapter,by discussing some properties of partitioned positive semi-definite matrix,polar decomposition,and trace and determinant inequalities of related matrix we obtained some new inequalities for determinants and traces of partitioned positive semi-definite matrix;then by studying the properties of nor-mal matrix and using singular value decomposition,we also obtained some new inequalities concerning traces,determinants and the spread of matrices.