Matrix scaling: A geometric proof of Sinkhorn's theorem
Let \(A\) be a real \(n \times n\) matrix whose entries are all positive. \textit{R. Sinkhorn} [Ann. Math. Stat. 35, 876-879 (1964; Zbl 0134.25302)] proved that there always exist diagonal matrices \(D_{1}\) and \(D_{2}\) with positive diagonal entries such that \(D_{1}AD_{2}\) is doubly stochastic. He did this by showing that the iterative process of alternately normalizing rows and columns converges. He also proved that the diagonal matrices \(D_{1}\) and \(D_{2}\) are uniquely determined up to the introduction of a positive scalar factor. Simple examples show that such diagonal matrices need not exist when \(A\) is only assumed to be nonnegative, but \textit{R. A. Brualdi, S. V. Parter} and \textit{H. Schneider} [J. Math. Anal. Appl. 16, 31-50 (1966; Zbl 0231.15017)] showed that Sinkhorn's theorem can be generalized to the case where \(A\) is nonnegative and fully indecomposable. The present paper gives a geometrical proof of the existence part of Sinkhorn's original theorem.
scientific article; zbMATH DE number 1100483
scientific article