A problem of Mirsky concerning nonsingular doubly stochastic matrices

The paper presents a new constructive proof of a theorem of Hardy, Littlewood, and Polya relating vector majorization and doubly stochastic matrices. Conditions on the vectors which guarantee that the corresponding matrices will be direct sums are given. These two results are applied to solve the problem, posed by Mirsky, of characterizing those majorization relations for which there is a corresponding doubly stochastic matrix which is nonsingular.