Linear maps preserving or strongly preserving majorization on matrices

‎For $A,B\in M_{nm},$ we say that $A$ is left matrix majorized (resp‎. ‎left matrix submajorized) by $B$ and write $A\prec_{\ell}B$ (resp‎. ‎$A\prec_{\ell s}B$)‎, ‎if $A=RB$ for some $n\times n$ row stochastic (resp‎. ‎row substochastic) matrix $R.$ Moreover‎, ‎we define the relation $\sim_{\ell s} $ on $M_{nm}$ as follows‎: ‎$A\sim_{\ell s} B$ if $A\prec_{\ell s} B\prec_{\ell s} A.$ This paper characterizes all linear preservers and all linear strong preservers of $\prec_{\ell s}$ and $\sim_{\ell s}$ from $M_{nm}$ to $M_{nm}$‎.