Stochastic matrices and a property of the infinite sequences of linear functionals

Our starting point is the proof of the following property of a particular class of matrices. Let T={Ti,j}T={Ti,j} be a n×mn×m non-negative matrix such that ∑jTi,j=1∑jTi,j=1 for each i  . Suppose that for every pair of indices (i,j)(i,j), there exists an index l   such that Ti,l≠Tj,lTi,l≠Tj,l. Then, there exists a real vector k=(k1,k2,…,km)T,ki≠kj,i≠j;0