A review of matrix scaling and Sinkhorn's normal form for matrices and positive maps

TL;DR

This paper reviews over 70 years of research on matrix scaling and Sinkhorn's theorem, emphasizing its mathematical foundations and recent generalizations to positive maps, with minimal new unpublished results.

Contribution

It provides a comprehensive overview of the mathematical landscape of matrix scaling and Sinkhorn's theorem, including recent extensions to positive maps.

Findings

Extensive historical overview of matrix scaling methods.

Discussion of generalizations to positive maps.

Highlights of mathematical techniques used in the field.

Abstract

Given a nonnegative matrix $A$, can you find diagonal matrices $D_1,~D_2$ such that $D_1AD_2$ is doubly stochastic? The answer to this question is known as Sinkhorn's theorem. It has been proved with a wide variety of methods, each presenting a variety of possible generalisations. Recently, generalisations such as to positive maps between matrix algebras have become more and more interesting for applications. This text gives a review of over 70 years of matrix scaling. The focus lies on the mathematical landscape surrounding the problem and its solution as well as the generalisation to positive maps and contains hardly any nontrivial unpublished results.