Linear Functions Preserving Multivariate and Directional Majorizat ion

Let $V$ and $W$ be two real vector spaces and let $\sim$ be a relation on both $V$ and $W$. A linear function $T:V \rightarrow W$ is said to be a linear preserver (respectively strong linear preserver) of $\sim$ if $Tx\sim Ty$ whenever $x\sim y$ (respectively $Tx\sim Ty$ if and only if $x\sim y$). In this paper we characterize all linear functions $T:\textbf{M}_{n,m} \rightarrow \textbf{M}_{n,k}$ which preserve or strongly preserve multivariate and directional majorization.