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Let $A$ be an nxm matrix.
We can easily determine its row and column marginals $r$ and $c$:
$r=A1$
$c=1^TA$.

Suppose however, that you are given non-negative marginals $r,c$.
Is there always a non-negative matrix $A$ whose marginals are $r$ and $c$ (obviously $r^T1 = 1^Tc$)?

tokumei
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  • What are your thoughts on the problem? Have you come up with any approaches to this question? Are you expecting the answer to be well known? – Ben Grossmann Nov 05 '14 at 04:41
  • @Omnomnomnom. I believe there should always be such a matrix A.
    Intuitively, for large enough A, the marginals induce an underdetermined system of linear equations. The only problem is the non-negativity.

    I suspect the result is known, or is implied by some other work on non-negative matrices. An approach I believe should work is using Farkas' Lemma. But, maybe there are simpler approaches...

     – tokumei Nov 05 '14 at 05:57
  • A similar question (see below) addresses the case where A has to have integer values. I am not interested in this special case, but the more general case. http://math.stackexchange.com/questions/315959/finding-all-possible-n-times-n-matrices-with-non-negative-entries-and-given-ro – tokumei Nov 05 '14 at 07:27

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