ISSN:
0945-3245
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract IfA is ann ×n matrix with strictly positive elements, then according to a theorem ofSinkhorn, there exist diagonal matricesD 1 andD 2 with strictly positive diagonal elements such thatD 1 A D 2 is doubly stochastic. This note offers an alternative proof of a generalization due toBrualdi, Parter andScheider, and independently toSinkhorn andKnopp, who show that A need not be strictly positive, but only fully indecomposable. In addition, we show that the same scaling is possible (withD 1 =D 2) whenA is strictly copositive, and also discuss related scaling for rectangular matrices. The proofs given show thatD 1 andD 2 can be obtained as the solution of an appropriate extremal problem. The scaled matrixD 1 A D 2 is of interest in connection with the problem of estimating the transition matrix of a Markov chain which is known to be doubly stochastic. The scaling may also be of interest as an aid in numerical computations.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF02170999
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