Matrix scaling: A geometric proof of Sinkhorn's theorem
type: Item
instance of: scholarly article
title: Matrix scaling: A geometric proof of Sinkhorn's theorem
zbMATH Open document ID: 0894.15008
DOI: 10.1016/S0024-3795(97)00010-4
author: Alberto Borobia
author: Rafael Cantó
published in: Linear Algebra and its Applications
publication date: 1998-08-12T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 15B51
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 15A12
zbMATH DE Number: 1100483
zbMATH Keywords: doubly stochastic matrix
zbMATH Keywords: matrix scaling
zbMATH Keywords: positive matrix
reviewed by: John D. Dixon
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1100483