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

Full-text PDF Free Access

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]**Richard A. Brualdi, Seymour V. Parter, and Hans Schneider,*The diagonal equivalence of a nonnegative matrix to a stochastic matrix*, J. Math. Anal. Appl.**16**(1966), 31–50. MR**0206019****[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**0215873**, 10.1090/S0002-9939-1967-0215873-6**[3]**Richard Sinkhorn,*A relationship between arbitrary positive matrices and doubly stochastic matrices*, Ann. Math. Statist.**35**(1964), 876–879. MR**0161868****[4]**Richard Sinkhorn and Paul Knopp,*Concerning nonnegative matrices and doubly stochastic matrices*, Pacific J. Math.**21**(1967), 343–348. MR**0210731**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
15.60,
65.00

Retrieve articles in all journals with MSC: 15.60, 65.00

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