A short note on doubly substochastic analog of Birkhoff's theorem

Abstract

Let B be an n by n doubly substochastic matrix. We show that B can be written as a convex combination of no more than {\sigma}(B)+t subpermutation matrices, where {\sigma}(B) is the number of nonzero elements in B and t is the number of fully indecomposable components of B^{comp}, the minimal doubly stochastic completion of B obtained by a specific way.