# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## Diagonal equivalence to matrices with prescribed row and column sums. IIHTML articles powered by AMS MathViewer

by Richard Sinkhorn
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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