Weak matrix majorization

Abstract

Given X, Y ∈ Rn×m we introduce the following notion of matrix majorization, called weak matrix majorization, X≻ w Y if there exists a row-stochastic matrix A ∈ Rn×n such that AX = Y, and consider the relations between this concept, strong majorization (≻s) and directional majorization (≻). It is verified that ≻ s ⇒ ≻w, but none of the reciprocal implications is true. Nevertheless, we study the implications ≻ w ⇒ ≻s and ≻ ⇒ ≻s under additional hypotheses. We give characterizations of strong, directional and weak matrix majorization in terms of convexity. We also introduce definitions for majorization between Abelian families of selfadjoint matrices, called joint majorizations. They are induced by the previously mentioned matrix majorizations. We obtain descriptions of these relations using convexity arguments.