On uniqueness, analyticity, and first- and second-order derivatives of Sinkhorn-type DAD scalings

We call a real square matrix A scalable if there is a diagonal matrix D with positive diagonal entries such that all row sums of D A D are equal to 1. In this case, Dis called a scaling for A. For an entrywise nonnegative A, this means that D A D is a stochastic matrix. We prove that such a nonnegative A has a unique scaling if and only if it has a scaling Dsuch that −1 is not an eigenvalue of D A D. If, on the contrary, a nonnegative A has multiple scalings, then we show that it already has infifinitely many scalings. Furthermore, we prove that the function which maps a uniquely scalable, nonnegative matrix Ato the diagonal vector x = x(A)of its scaling, which we call the Sinkhorn vector of A, is a real analytic function. Finally, we give explicit, index-free formulas for the Jacobian and Hessian matrices of xwith respect to the entries of A. In particular, we prove that ∂xi/∂Ai,j ≤ −1/2xi²xj < 0 for all i, j ∈ {1,...,n} where n is the order of A.