Linear converters of majorization to row majorization and vice versa

Let Rn be the set of all real vectors. For every x,y∈Rn it is said that x is majorized by y (denoted by x≺y) if there exists a doubly stochastic matrix D such that x = Dy. Also it is said that x is row majorized by y (denoted by x≺ry) if there exists a row stochastic matrix R such that x = Ry. So far, the forms of linear operators T:Rn⟶Rn such that Tx≺Ty whenever x≺y and Tx≺rTy whenever x≺ry have been obtained. We say that a linear operator T:Rn→Rn is a linear converter of majorization to row majorization if Tx≺rTy whenever x≺y and a linear converter of row majorization to majorization if Tx≺Ty whenever x≺ry. In this paper we find the structure of all linear converters of majorization to row majorization on Rn and vice versa.