Vector majorization via nonnegative definite doubly stochastic matrices of maximum rank

It is known that for real n-vectors x and y, y majorizes x if and only if Ay = x for some doubly stochastic matrix A of order n. Suppose that the coordinates of each of x and y are in nonincreasing order. Then the matrix A can be chosen to be nonnegative definite. If there is no coincidence and if the coordinates of x are not all equal, then A can be chosen to be positive definite. In this paper, we obtain a simple formula for calculating the maximum rank of all nonnegative definite doubly stochastic matrices A such that Ay = x.