Scaling of Matrices to Achieve Specified Column and Row Sums

摘要

If A is an n x n matrix with strictly positive elements, then according to a theorem of Sinkhorn, there exist diagonal matrices D sub 1 and D sub 2 with strictly positive diagonal elements such that D1AD2 is doubly stochastic. This note offers an alternative proof of a generalization due to Brualdi, Parter and Schneider, and independently to Sinkhorn and Knopp, who show that A need not be strictly positive, but only fully indecomposable. In addition, we show that the same scaling is possible (with D sub 1 = D sub 2) when A is strictly copositive, and also discuss related scaling for rectangular matrices. The proofs given show that D sub 1 and D sub 2 can be obtained as the solution of an appropriate extremal problem. The scaled matrix D1AD2 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. (Author)