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 be a nonnegative square matrix and where and are diagonal matrices with positive diagonal entries. Several proofs are known for the following theorem: If is fully indecomposable then and can be chosen so that is doubly stochastic. Moreover, and 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.

**[1]**R. A. Brualdi, S. V. Parter and H. Schneider,*The diagonal equivalence of a nonnegative matrix to a stochastic matrix*, J. Math. Anal. Appl.**16**(1966), 31-50. MR**34**#5844. MR**0206019 (34:5844)****[2]**M. V. Menon,*Reduction of a matrix with positive elements to a doubly stochastic matrix*, Proc. Amer. Math. Soc.**18**(1967), 244-247. MR**35**#6708. MR**0215873 (35:6708)****[3]**R. Sinkhorn,*A relationship between arbitrary positive matrices and doubly stochastic matrices*, Ann. Math. Statist.**35**(1964), 876-879. MR**28**#5072. MR**0161868 (28:5072)****[4]**R. Sinkhorn and P. Knopp,*Concerning nonnegative matrices and doubly stochastic matrices*, Pacific J. Math.**21**(1967), 343-348. MR**35**#1617. MR**0210731 (35:1617)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1970-0257114-X

Keywords:
Nonnegative matrix,
doubly stochastic matrix,
irreducible matrix,
fully indecomposable matrix

Article copyright:
© Copyright 1970
American Mathematical Society