Some Results on Majorization of Matrices

For two $n\times m$ real matrices X and Y, X is said to be majorized by Y, written as $X\prec Y$ 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 $X\prec Y$ 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 $X\prec Y$ for the cases where the rank of $Y=n-2$ and in general for rank of $Y=n-k$, where $1\le k\le n-1$. 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 $S\in {\mathsf{\Omega }}_m$ 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 $X{⪯}^{c}Y,$ 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.