Suppose we have an n by m nonnegative matrix A, where each row sums to 1. I wonder whether there exists an m by n nonnegative matrix X that satisfies the following constraints: each row of X sums to 1; the diagonal entries of matrix AX are all equal and they are lager or equal to any other entries of the matrix.

If the solution does not always exist, would like to know iff conditions for such a solution to exist.

There are existence and uniqueness results for linear equations here and there, but my problem is the existence of a matrix with some particular constraints. Has been a while and still have no clue how to attack it.