Linear Preservers of Doubly stochastic matrices and permutation matrices from $M_m$ to $M_n$

Chi-Kwang Li, Bit-Shun Tam and Nam-Kiu Tsing have obtained necessary and sufficient condition for a linear operator on linear space of generalized doubly stochastic matrices to be strong preserver of doubly stochastic matrices and permutation matrices. We show if a linear operator T : Mm → Mn is a (strong) preserver of doubly stochastic matrices, then T is a (strong) preserver of the linear manifold of r-generalized doubly stochastic matrices and the linear space of generalized doubly stochastic matrices. Also we give necessary and sufficient condition for a linear operator T : Mm → Mn to be (strong) preserver of doubly stochastic matrices and permutation matrices.