Published online by Cambridge University Press: 03 November 2016
A square matrix is said to be doubly-stochastic if its elements are non-negative and if all row-sums and column-sums are equal to 1. The study of doubly-stochastic matrices was initiated by I. Schur and was subsequently taken up by Hardy, Littlewood, and Polya, who proved the following fundamental proposition [1, Theorem 46].