A note on multivariate majorization

𝐴 matrix 𝐴 is said to be multivariate majorized by a matrix 𝐵, written 𝐴 𝐵, if there exists a doubly stochastic matrix 𝐷 such that 𝐴 = 𝐵𝐷 . In the present paper, we obtain a totally ordered subset of Mnm which contains a given matrix 𝐴. Also, we show that the totality of all extreme points of the collection of all matrices which are multivariately majorized by a matrix 𝐴 is the set of all matrices obtained by permuting the columns of 𝐴.