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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 knownas Sinkhorn's theorem. It has been proved with a wide variety of methods, eachpresenting a variety of possible generalisations. Recently, generalisationssuch as to positive maps between matrix algebras have become more and moreinteresting for applications. This text gives a review of over 70 years ofmatrix scaling. The focus lies on the mathematical landscape surrounding theproblem and its solution as well as the generalisation to positive maps andcontains hardly any nontrivial unpublished results.

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