Forward Order Laws For Generalized Inverse Of Multiple Matrix Products

It is well know that(A1A2…An)-1 =An-1An-1-1…A1-1 holds for the product of m nonsingular matrices Ai(i = 1,2,… n).In recent years,many readers studied and given the necessary and sufficient conditions for one side inclusion An{i,j,k}An-1{i,j,k}… A1{i,j,k}(?)(A1A2 … An){i,j,k}.As one of the core problems in generalized inverse,the sufficient and necessary conditions for the forward order laws for geeneralized inverse of matrix product hold is the fundamental in the theory of generalized inverse of matrix,which has attracted considerable attention.In this paper,by using the extreme ranks of the generalized Schur complement,we will give the necessary and sufficient conditions of the forward order laws for the generalized inverse of multiple matrix products:A1{i,j,k}A2{i,j,k]…An{ij,k}(?)(A1A2…An){i,j,k),where Ai{i,j,k)denotes the set of {i,j,k}-inverse of Ai and Ai ? Cm×n(i = 1,2… n).The related paper structure is as follows:In Chapter 1,we mainly introduces the research background,the significance of the research,and the research progress of domestic and foreign counterparts at the present stage.In Chapter 2,we mainly introduces the basic concepts,basic properties,basic theorems and basic definitions of the knowledge involved in the paper.In Chapter 3 investigstes the necessary and sufficient conditions of the forward order laws for generalized inverses of shree matrix products.In Chapter 4 investigstes the necessary and sufficient conditions for the following forward order laws:A1{1,3}A2{1,3} …An{1,3}(?)(A1A2 …An){1,3},A1{1,4}A2{1,4} …An{1,4}(?)(A1A2 …An){1,4},A1{1,2,3}A2{1,2,3} …An{1,2,3}(?)(A1A2 …An){1,2,3},A1{1,2,4}A2{1,2,4} …An{1,2,4}(?)(A1A2 …An){1,2,4}.