Research On The Properties Of Matrix Partial Order And EP Matrix

Matrix partial order and EP matrix play an important role in matrix theory,matrix partial order not only enriches the content of this subject,but also promotes the progress of many other disciplines.It is an indispensable mathematical tool in many fields,such as modern natural science,social science and even engineering technology.EP matrices are a special class of matrices,which are closely related to many other special matrices.Based on the study of matrix theory by experts and scholars,this paper st.arts with the basic definitions and matrix decompositions,and then explores the properties of matrix partial orders,the relations of several partial orders,the relations between EP matrix and other special matrices,and the partial orders properties of EP matrix.This paper is divided into three chapters,the main contents are as follows:In the first chapter,we introduce some basic definitions and lemmas used in this paper.Such as:the definitions of 1-inverse of matrix,index,Moore-Penrose inverse,group inverse,Drazin inverse,core inverse,twelve kinds of special matrices,five kinds of partial orders,two kinds of preorderings,as well as the related lemmas used in this paper.In the second chapter,we study the problems of matrix partial orders.We discuss the related properties of the five kinds of partial orders and two kinds of preorderings by the tools of decompositions and the generalized inverses of matrices.We study the partial orders relations between general complex matrix and their squares in the sense of minus partial order,star partial order,sharp partial order,core partial order and Drazin order.Secondly,the complex matrix is transformed into Hermite positive semi-definite matrix,we obtain the partial orders relations between Hermite positive semi-definite matrix and their squares in the sense of Lowner partial order,minus partial order,star partial order.Lastly,we study the relations among minus partial order,star partial order,sharp partial order and core partial order.We obtain the relations among idempotent matrix space order,minus partial order and sharp partial order by use the full rank decomposition of matrix.In the third chapter,we use the definitions of EP matrix and Hartwig decom-position to study the related problems of EP matrix.Firstly,the properties of EP matrix are deeply characterized,the equivalent propositions of EP matrix,the rela-tions between EP matrix and several special matrices,and the closure of EP matrix are obtained.Secondly,with combine EP matrix and partial orders,we obtain the relations among minus partial order,star partial order,sharp partial order,core partial order and Drazin order.