A minimal completion of doubly substochastic matrix

Partitions of the polytope of doubly substochastic matrices

“…Proof. (i) According to Theorem 2.2 [7], if B ∈ ω n,k , then σ(B) is in [n − k, n − k + 1) for 1 ≤ k ≤ n. Therefore,…”

“…The first way to partition ω n is induced by a characteristic of doubly substochastic matrices called sub-defect [7,6,5]. Definition 2.1.…”

“…Definition 2.1. ( [7]) The sub-defect of an n × n doubly substochastic matrix B, denoted by sd(B), is defined to be the smallest integer k such that there exists an (n + k) × (n + k) doubly stochastic matrix containing B as a submatrix.…”

Sub-defect of product of doubly substochastic matrices

“…Given a doubly substochastic matrix, a natural question is what is the minimal size of a doubly stochastic matrix which contains the given doubly substochastic matrix as a submatrix. This question was addressed and answered by the authors of [5] in which an interesting characteristic, sub-defect, of doubly substochastic matrices was defined and investigated. Definition 1: The sub-defect of an n × n doubly substochastic matrix B, denoted by sd(B), is defined to be the smallest integer k such that there exists an (n + k) × (n + k) doubly stochastic matrix containing B as a submatrix.…”

“…The main contribution of [5] is that the sub-defect of any doubly substochastic matrix can be calculated by taking the ceiling of the difference of its size and the sum of all entries. Let n denote the set of doubly stochastic matrices, ω n denote the set of doubly substochastic matrices and ω n,k denote the set of all doubly substochastic matrices with sub-defect k. We note that the sub-defect provides a way to partition the set ω n into n + 1 convex subsets ω n,0 = n , ω n,1 , .…”