Weak matrix majorization Original Research Article

Given image we introduce the following notion of matrix majorization, called weak matrix majorization,imageand consider the relations between this concept, strong majorization (succeedss) and directional majorization (succeeds). It is verified that succeedss implies succeeds implies succeedsw, but none of the reciprocal implications is true. Nevertheless, we study the implications succeedsw implies succeedss and succeeds implies succeedss 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.