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Proceedings of the American Mathematical Society

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



Note on nonnegative matrices

Author: D. Ž. Djoković
Journal: Proc. Amer. Math. Soc. 25 (1970), 80-82
MSC: Primary 15.60; Secondary 65.00
MathSciNet review: 0257114
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Abstract: Let $A$ be a nonnegative square matrix and $B = {D_1}A{D_2}$ where ${D_1}$ and ${D_2}$ are diagonal matrices with positive diagonal entries. Several proofs are known for the following theorem: If $A$ is fully indecomposable then ${D_1}$ and ${D_2}$ can be chosen so that $B$ is doubly stochastic. Moreover, ${D_1}$ and ${D_2}$ are unique up to a scalar factor. It is shown that these results can be easily obtained by considering a minimum of a certain rational function of several variables.

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Keywords: Nonnegative matrix, doubly stochastic matrix, irreducible matrix, fully indecomposable matrix
Article copyright: © Copyright 1970 American Mathematical Society