Joint majorization

Given A;B ∈ Mnm(C) we introduce the following notion of majorization A ≺g B if there exists a generalised doubly stochastic matrix D ∈ Mn(C) such that A = DB; and consider the majorization on ma- trices which is denoted by A ⪯ B: A ⪯ B whenever λ(A) is multivariat majorized by λ(B) i.e. λ(A) = Dλ(B) for some doubly stochastic matrix D: We give a characterization of ⪯ based on convexity. We also introduce majorization between two n ✖ n abelian diagonalizable matrices (ai)i=1;... ;m and (bi)i=1;... ;m which is called joint majorization. Also, we give a characterization of joint majorization when abelian families are normal matrices