Matrix majorization via vector majorization

For two real m×n matrices X and Y, Y is said to majorize X if SY=X for some doubly stochastic matrix S of order m. In this note, we prove that Y majorizes X if and only if Yv majorizes Xv for every real n-vector v, under the assumption that [X,e][Y,e]+ is nonnegative, where e and [Y,e]+ denote the m-vector of ones and the Moore–Penrose generalized inverse of [Y,e], respectively.