Sg-Majorization on R2

Let Mn be the vector space of all n-by-n real matrices. A (not necessarily nonnegative) matrix R ∈ Mn with the property Re = e is said to be a g-row stochastic matrix, where e = (1,..., 1)t. For x, y ∈ Rn , we say that x sg-majorized by y (write as x ≺sg y) if for some symmetric g-row stochastic matrix R with all its main diagonal entries equal, x = Ry. Here, we give an equivalent condition for sg-majorization on R2. Also, the possible structures of linear preserving sg-majorization functions from R2 to R2 are found. Finally, all linear strongly preserving ≺sg from R2 to R2 are characterized.