Suppose we have an $n \times m$ nonnegative matrix $A$, where each row sums to $1$. I wonder whether there exists an $m \times n$ 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 larger or equal to any other entries of the matrix.

If the solution does not always exist, would be interesting to know necessary and sufficient conditions for such a solution to exist.

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