Showing papers by "Ingram Olkin published in 1968"
TL;DR: In this paper, it was shown that if A is an ann × n matrix with strictly positive elements, then according to a theorem ofSinkhorn, there exist diagonal matrices D 1 and D 2 with strictly-positive diagonal elements such that D 1 A D 2 is doubly stochastic.
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.
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