Matrix scaling and explicit doubly stochastic limits

The process of alternately row scaling and column scaling a positive $n \times n$ matrix $A$ converges to a doubly stochastic positive $n \times n$ matrix $S(A)$, often called the \emph{Sinkhorn limit} of $A$. The main result in this paper is the computation of exact formulae for the Sinkhorn limits of certain symmetric positive $3\times 3$ matrices.