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The alternate row and column scaling algorithm applied to a positive $n\timesn$ matrix $A$ converges to a doubly stochastic matrix $S(A)$, sometimes calledthe \emph{Sinkhorn limit} of $A$. For every positive integer $n$, a twoparameter family of row but not column stochastic $n\times n$ positive matricesis constructed that become doubly stochastic after exactly one column scaling.

Matrix scaling limits in finitely many iterations