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:M_m\rightarrow M_n 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:M_m\rightarrow M_n to be (strong) preserver of doubly stochastic matrices and permutation matrices.

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ماتریسهای تصادفی دوگانه , حافظ خطی , منیفلدهای خطی