Note on nonnegative matrices
Author:
D. Ž. Djoković
Journal:
Proc. Amer. Math. Soc. 25 (1970), 80-82
MSC:
Primary 15.60; Secondary 65.00
DOI:
https://doi.org/10.1090/S0002-9939-1970-0257114-X
MathSciNet review:
0257114
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Abstract | References | Similar Articles | Additional Information
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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- M. V. Menon, Reduction of a matrix with positive elements to a doubly stochastic matrix, Proc. Amer. Math. Soc. 18 (1967), 244–247. MR 215873, DOI https://doi.org/10.1090/S0002-9939-1967-0215873-6
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Additional Information
Keywords:
Nonnegative matrix,
doubly stochastic matrix,
irreducible matrix,
fully indecomposable matrix
Article copyright:
© Copyright 1970
American Mathematical Society