Study On Local Features Of Linear Mappings Preserving Majorization

In the research of majorization,the linear mapping preserving majorization is a quite interesting direction.In recent years,people research various types of linear mappings preserving majorization,such as strong linear preservers of multivariate majorization,linear mappings preserving local multivariate majorization,linear maps preserving G-majorization,linear maps preserving row and column-majorization,linear preservers of B-majorization so on.And there have been abundant results,for example,Ando char-acterized the form of linear mappings preserving majorization in vector space;Li and Poon showed that the linear operators preserving directional majorization is equivalent to it preserving multivariate majorization.This thesis studies the local feature of linear mapping preserving majorization on R3 and Mn,m,it organized as follows:In chapter 1,it introduces the context of majorization and the current research direc-tion of linear mappings preserving majorization.In addition,it gives the description of some symbols used in this thesis.In chapter 2,it gives the concept of strong local linear preservers of matrix majoriza-tion of type I and strong local linear preservers of matrix majorization of type II from the strong linear preservers of matrix majorization and the local linear preservers of matrix majorization.It characterizes the form and properties of the two types of the strong local linear preservers of matrix majorization.In addition,it shows that the strong local linear preservers of matrix majorization of type I is the strong local linear preservers of matrix majorization of type II.In chapter 3,by using classification and reduction to absurdity,it proves that the linear mapping of right preserving-majorization at the circulant points on R3 with different components,whose sum is not equal to 0,is equivalent to the linear mapping preserving majorization.Also,the linear mapping of left preserving-majorization at the circulant points with only two same components,whose sum is not equal to 0,is equivalent to the linear mapping preserving majorization.In chapter 4,it gives a summary of this thesis and proposes some future study direc-tions.