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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Author: Richard Sinkhorn
Journal: Proc. Amer. Math. Soc. 45 (1974), 195-198
MSC: Primary 15A21
MathSciNet review: 0357434
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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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1974-0357434-8
PII: S 0002-9939(1974)0357434-8
Keywords: Nonnegative matrix, diagonal equivalence, fully indecomposable matrix, zero pattern
Article copyright: © Copyright 1974 American Mathematical Society