Suppose $r_i$, $1 \le i \le n$, and $c_j$, $1 \le j \le m$, are nonnegative integers. When does there exist an $n \times m$ matrix in $\text{Mat}_{n \times m} (\mathbb{Z}^+)$, i.e. nonnegative entries, such that $r_i$ is the sum of the entries in its $i$th row and $c_j$ is the sum of the entries in its $j$th columns?
Constraints on sum of rows and columns of matrix
4
$\begingroup$
combinatorics
algebra-precalculus
matrices
-
1@user314: Why the algebra-precalculus tag? – 2010-11-16
1 Answers
6
Non-negative integer solutions exist whenever the (also necessary) "balance condition" is met:
$$ \sum_i r_i = \sum_j c_j $$
which simply sums the total matrix entries in two ways.
There is a survey paper by Alexander Barvinok on this: http://www.math.lsa.umich.edu/~barvinok/linalg.pdf
-
1Yes, it's a very active area. One of my fellow graduate students is working on this for his thesis. – 2010-11-16