The doubly stochastic matrices of a vector majorization
Let x and y be real n-dimensional column vectors; y is said to be majorized by x if the following holds: if \(x_{[1]},...,x_{[n]}\) are the components of x arranged in nonincreasing order and \(y_{[1]},...,y_{[n]}\) are likewise for y, then \(\sum^{k}_{i=1}y_{[i]}\leq \sum^{k}_{i=1}x_{[i]},\) \(k=1,...,n\), with equality for \(k=n\). The polytope of doubly stochastic matrices D for which \(y=Dx\) is investigated. It is determined when there exists a positive D and when there exists a fully indecomposable D. The dimension of the polytope is computed and as a consequence it is determined just when D is unique.
scientific article; zbMATH DE number 3889725