Can every doubly stochastic be generated by the Sinkhorn Knopp algorithm?

Question

I want to generate doubly stochastic matrices in a way that doesn't exclude any doubly stochastic matrix. At the moment I am generating an $n \times n$ matrix whose components are from a $\text{Uniform}(0,1)$, then normalising it using the Sinkhorn Knopp algorithm to end up with a doubly stochastic matrix. My question is, does this method of generating doubly stochastic matrices exclude the possibility of returning any doubly stochastic matrices? I'm quite certain it doesn't, but how can I show this?

Answer 1

If the initial matrix you generate is already doubly stochastic, the SK algorithm will leave it unchanged. So any doubly stochastic matrix is a possible result of the method.