A minimal completion of double substochastic matrices

Abstract

Let B be an n x n doubly substochastic matrix and let s be the sum of all entries of B. In this paper we show that B has a sub-defect of k which can be computed by taking the ceiling of (n-s) if and only if there exists an (n+k) x (n+k) doubly stochastic extension containing B as a submatrix and k minimal. We also propose a procedure constructing a minimal completion of B, and then express it as a convex combination of partial permutation matrices.