doubly stochastic matrices

T. Ando characterized linear preservers of majorization in  [Linear Algebra Appl. 118 (1989) 163-248]. In this note, we present a method to state a simple proof of Ando's theorem. By using this method, we state an equivalent condition for matrix representations of linear preservers of $G$-majorization on matrices, where  $G$ is a finite subgroup of orthogonal  group $O(\mathbb{R}^n)$.Moreover, we introduce reflective majorization and characterize all its linear preservers.