A note on multivariate majorization

Abstract

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

Keywords

• doubly stochastic‎
• ‎extreme point
• ‎multivariate majorization