Decomposability of Weak Majorization

Let x; y 2 Rn: We use the notation x w y when x is weakly majorized by y. We say that x w y is decomposable at k (1 k < n) if x w y has a coincidence at k and yk , yk+1. Corresponding to this majorization we have a doubly substochastic matrix P. The paper presents x w y is decomposable at some k (1 k < n) if and only if P is of the form D Q where D and Q are doubly stochastic and doubly substochastic matrices, respectively. Also, we write some algorithms to obtain x from y when x w y.