Some Results on Majorization of Matrices

For two n×m real matrices X and Y, X is said to be majorized by Y, written as X≺Y if X=SY for some doubly stochastic matrix of order n. Matrix majorization has several applications in statistics, wireless communications and other fields of science and engineering. Hwang and Park obtained the necessary and sufficient conditions for X,Y to satisfy X≺Y for the cases where the rank of Y=n−1 and the rank of Y=n. In this paper, we obtain some necessary and sufficient conditions for X,Y to satisfy X≺Y for the cases where the rank of Y=n−2 and in general for rank of Y=n−k, where 1≤k≤n−1. We obtain some necessary and sufficient conditions for X to be majorized by Y with some conditions on X and Y. The matrix X is said to be doubly stochastic majorized by Y if there is S∈Ωm such that X=YS. In this paper, we obtain some necessary and sufficient conditions for X to be doubly stochastic majorized by Y. We introduced a new concept of column stochastic majorization in this paper. A matrix X is said to be column stochastic majorized by Y, denoted as X⪯cY, if there exists a column stochastic matrix S such that X=SY. We give characterizations of column stochastic majorization and doubly stochastic majorization for (0,1) matrices.