The Properties Of Two Kinds Of Special Matrices And The Study Of The Rank Disturbance Of The 2×2 Partitioned Matrix

The theory of linear algebra plays an important role in many fields.Its research has been lasted for more than a century.Matrix theory has been widely used in applied mathematics,computer science,economics,engineering science,operations research,statistics and other disciplines.Matrix theory is indispensable for combi-natorial mathematics,group theory,graph theory,operator theory and other math disciplines.It is one of the most abundant disciplines in mathematics.Based on the results of the previous research,this paper research EP matrix,normal matrix and rank disturbance of the 2×2 partitioned matrix.Some important conclusions are drawn.This paper is divided into four chapters as follows:In chapter one,some fundamental definition,theorem are introduced,including generalized inverse matrix,Moore-Penrose inverse,Drazin inverse,group inverse,Singular Value Decomposition,EP matrix and equivalent conditions of normal ma-trix.In chapter two,EP matrix is a special and important matrix,which is related to normal matrix.Using the basic definition of EP matrix,Moore-Penrose inverse and the properties of EP matrix,we obtain some new properties and inferences of EP matrix.Next,the special matrix decomposition is constructed and the equivalent conditions of EP matrix are got.In chapter three,normal matrix is a special and important matrix,and ap-plied in many fields.Firstly,the special matrix decomposition is used,the equiv-alent conditions of normal matrix are got.Secondly,by using previous studies,some important conclusions of normal matrix,Unitary matrix,Hermite matrix and Skew-Hermite matrix are obtained.Finally,by using mathematical induction,the inequalities of normal matrix are obtained.In chapter four,rank of matrix is an important quantitative index of matrix and plays an important role in matrix theory.It is indispensable for system of linear equations,quadratic form,linear dependence of vector group,generalized inverse matrix and other algebra questions and accordingly the rank of matrix is always a research hotspot.By using the skill of matrix Schur complement and the characteristic of generalized inverse matrix,properties and rank disturbance of partitioned matrix are researched,invariance property of partitioned matrix with parameters and rank of matrix function are discussed,the relationship between upper and lower bound of rank of partitioned matrix and disturbance factor are obtained,the results of rank of partitioned matrix are deepened and the theory basis for the research on rank reduction partial order are provided.