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

 

Nonnegative matrices each of whose positive diagonals has the same sum


Author: Mark Blondeau Hedrick
Journal: Proc. Amer. Math. Soc. 59 (1976), 399-403
MSC: Primary 15A48
MathSciNet review: 0414595
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Abstract: The author shows that if $ A$ is a fully indecomposable nonnegative matrix each of whose positive diagonals has sum $ M$ and when $ {a_{ij}} = 0$, the sum of each positive diagonal in the submatrix of $ A$ obtained by deleting the $ i{\text{th}}$ row and $ j{\text{th}}$ column is less than $ M$, then there is a unique positive matrix $ B$ such that its rank is at most two, each of its diagonals has sum $ M$, and $ {a_{ij}} = {b_{ij}}$ when $ {a_{ij}} > 0$. The author then compares his results to those obtained by Sinkhorn and Knopp who carried out a similar analysis for positive diagonal products.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1976-0414595-1
PII: S 0002-9939(1976)0414595-1
Keywords: Fully indecomposable, doubly stochastic pattern, elementary symmetric function of a diagonal
Article copyright: © Copyright 1976 American Mathematical Society