On Sinkhorn's <i>DAD</i> theorem and the self-consistency equation in COSMO-based activity coefficient models

In a 1966 paper, Sinkhorn proved that for any real square matrix <i>A</i> which has only positive entries there exists a uniquely determined real diagonal matrix <i>D</i> with positive diagonal entries such that B:=DAD is stochastic, i.e. all row sums of <i>B</i> are equal to 1. Moreover, Sinkhorn stated an iterative method for computing <i>D</i>. Nowadays, Sinkhorn's result and its variants are often referred to as <i>DAD</i> theorems. The purpose of this article is twofold. On the one hand, we give the link between Sinkhorn's <i>DAD</i> theorem and the self-consistency equation in COSMO-based activity coefficient models in chemical engineering. On the other hand, we give a new constructive proof of Sinkhorn's <i>DAD</i> theorem by using classical fixed-point theory. Hereby, the larger class of nonnegative matrices with positive diagonal is considered. Our proof uniformly provides convergence for a number of iterative methods for computing <i>D</i>. Some of them are used in practice although, to the best of our knowledge, a formal proof of convergence is missing.