Diagonal Sums of Doubly Substochastic Matrices

Let Ω_n denote the convex polytope of all n x n doubly stochastic matrices, and ω_n denote the convex polytope of all n x n doubly substochastic matrices. For a matrix A ϵ ω_n, define the sub-defect of A to be the smallest integer k such that there exists an (n + k) x (n + k) doubly stochastic matrix containing A as a submatrix. Let ω_n,k denote the subset of ω_n which contains all doubly substochastic matrices with sub-defect k. For π a permutation of symmetric group of degree n, the sequence of elements a_1π(1); a_2π(2), ..., a_nπ(n) is called the diagonal of A corresponding to π. Let h(A) and l(A) denote the maximum and minimum diagonal sums of A ϵ ω_n,k, respectively. In this paper, existing results of h and l functions are extended from Ω_n to ω_n,k. In addition, an analogue of Sylvesters law of the h function on ω_n,k is proved.