Linear preserving gd-majorization functions from Mn,m to Mn,k

Let $\mathbf{M}_{n,m}$ be the vector space of all $n\times‎ ‎m$ real matrices‎. ‎For $A,B\in \mathbf{M}_{n,m}$‎, ‎it is said that‎ ‎\textit{B} is \textit{gd-majorized} by \textit{A} (written‎ ‎$A\succ_{gd}B$) if for every $x\in \mathbb{R}^n$ there exists a‎ ‎g-doubly stochastic matrix $D_x$ such that $Bx=D_x(Ax)$‎. ‎Here‎, ‎we‎ ‎show that if $A\succ_{gd}B$‎, ‎then there exists a g-doubly‎ ‎stochastic matrix $D$ (independent of $x$) such that $B=DA$‎. ‎Also‎, ‎the possible structures of linear preserving‎ ‎gd-majorization functions from $ \mathbf{M}_{n,m}$ to $‎ ‎\mathbf{M}_{n,k}$ are found‎. ‎Finally‎, ‎all linear strongly‎ ‎preserving gd-majorization functions from $ \mathbf{M}_{n,m}$ to $‎ ‎\mathbf{M}_{n,k}$ are characterized‎.