Proofs of two theorems on doubly-stochastic matrices

Abstract

here xl, * * *, xn denote the numbers x1, , xn arranged in nonascending order of magnitude, and similarly for the y's. All matrices we consider are of type nXn, unless the contrary is obvious from the context; a matrix is called doubly-stochastic (d.s.) if its elements are non-negative numbers such that the sum of elements in each row and in each column is equal to 1. The set of all nXn d.s. matrices will be denoted by )n. Further, we denote by (En the set of the n! permutations of 1, , n; and by Wo the identical permutation in (.n H(xi, * * *, xn) is the convex hull of the n! vectors (x,i, ,n) 7r-n. Finally, 8kj denotes the Kronecker delta. The object of the present note is to give simple proofs of two results on d.s. matrices. The first of these results is due to Hardy, Littlewood, and Polya ([3, Theorem 46]; for an alternative proof see [5, ??5-8]); the second is due to G. Birkhoff [2, ?1].