Convex Set of Doubly Substochastic Matrices

Abstract

Denote $\mathcal{A}$ as the set of all doubly substochastic m×n matrices and let k be a positive integer. Let $\mathcal{A}_k$ be the set of all 1/k-bounded doubly substochastic m × n matrices, i.e., $\mathcal{A}_k \triangleq \{E \in \mathcal{A}:e_{i,j} \in [0,1/k],\forall i = 1,2, \cdots ,m,j = 1,2, \cdots ,n\}$. Denote ℬk as the set of all matrices in $\mathcal{A}_k$ whose entries are either 0 or 1/k. We prove that $\mathcal{A}_k$ is the convex hull of all matrices in ℬk. In addition, we introduce an application of this result in communication system.