A closed form for the 2x2 generalized Sinkhorn limit is presented, together with an incomplete proof sketch that all generalized Sinkhorn entries are algebraic with bounded degree.
Alternate minimization and doubly stochastic matrices
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abstract
Sinkhorn's alternative minimization algorithm applied to a positive $n\times n$ matrix converges to a doubly stochastic matrix. If the algorithm, applied to a $2\times 2$ matrix, converges in a finite number of iterations, then it converges in at most two iterations, and the structure of such matrices is determined.
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Closed Form of a Generalized Sinkhorn Limit
A closed form for the 2x2 generalized Sinkhorn limit is presented, together with an incomplete proof sketch that all generalized Sinkhorn entries are algebraic with bounded degree.