เรื่อง Sinkhorn limits of 3x3 symmetric matrices and their doubly stochastic shapes
Let A and B be positive nxn matrices. We write $A \sim B$ if there exist nxn
permutation matrices P and Q and $\lambda > 0$ such that
$B =\lambda PAQ$
This is an equivalence relation. Moreover, $A \sim B$ implies
$S(B) = \lambda PS(A)Q$
Thus, it suffices to determine the Sinkhorn limit of only one matrix in an equivalence class.
We shall compute the Sinkhorn limit of every symmetric positive 3x3 matrix
whose set of coordinates consists of two distinct real numbers.
Let A be such a matrix with coordinates M and N with M = N. There are 9
coordinate positions in the matrix, and so exactly one of the numbers M and N
occurs at least five times. Suppose that the coordinate M occurs five or more times.
Let $\lambda = 1/M$ and $K = N/M$ The matrix $\lambda A$ has two distinct positive coordinates
1 and K, and K occurs at most four times. There are seven equivalence classes of
such matrices with respect to permutations and dilations. The main result of this
paper is the calculation of the Sinkhorn limits of these matrices.
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https://arxiv.org/pdf/1905.09426.pdf