Diagonal equivalence to matrices with prescribed row and column sums. II

Sinkhorn, Richard

doi:10.1090/S0002-9939-1974-0357434-8

Diagonal equivalence to matrices with prescribed row and column sums. II
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Proc. Amer. Math. Soc. 45 (1974), 195-198 Request permission

Abstract:

Let $A$ be a nonnegative $m \times n$ matrix and let $r = ({r_1}, \cdots ,{r_m})$ and $c = ({c_1}, \cdots ,{c_n})$ be positive vectors such that $\Sigma _{i = 1}^m{r_i} = \Sigma _{j = 1}^n{c_j}$. It is well known that if there exists a nonnegative $m \times n$ matrix $B$ with the same zero pattern as $A$ having the $i$th row sum ${r_i}$ and $j$th column sum ${c_j}$, there exist diagonal matrices ${D_1}$ and ${D_2}$ with positive main diagonals such that ${D_1}A{D_2}$ has $i$th row sum ${r_i}$ and $j$th column sum ${c_j}$. However the known proofs are at best cumbersome. It is shown here that this result can be obtained by considering the minimum of a certain real-valued function of $n$ positive variables.

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