The structure of strong linear preservers of gw-majorization on $ \mathbf{M}_{{n,m}

Let M_{n,m} be the set of all n by m matrices with entries in F, where F is the field of real or complex numbers. A matrix R in M_{n} with the property Re=e, is said to be a g-row stochastic (generalized row stochastic) matrix. Let A,B in M_{n,m}, so B is said to be gw-majorized by A if there exists an n by n g-row stochastic matrix R such that B=RA. In this paper we characterize all linear operators that strongly preserve gw-majorization on M_{n,m} and all linear operators that strongly preserve matrix majorization on M_{n} .