Note on nonnegative matrices
HTML articles powered by AMS MathViewer
- by D. Ž. Djoković
- Proc. Amer. Math. Soc. 25 (1970), 80-82
- DOI: https://doi.org/10.1090/S0002-9939-1970-0257114-X
- PDF | Request permission
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.References
- 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 206019, DOI 10.1016/0022-247X(66)90184-3
- 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 10.1090/S0002-9939-1967-0215873-6
- Richard Sinkhorn, A relationship between arbitrary positive matrices and doubly stochastic matrices, Ann. Math. Statist. 35 (1964), 876–879. MR 161868, DOI 10.1214/aoms/1177703591
- Richard Sinkhorn and Paul Knopp, Concerning nonnegative matrices and doubly stochastic matrices, Pacific J. Math. 21 (1967), 343–348. MR 210731
Bibliographic Information
- © Copyright 1970 American Mathematical Society
- 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