Electronic Journal of Linear Algebra
Doubly substochastic matrices, Sub-defect, Maximum diagonal sum
Let Ωn denote the convex polytope of all n x n doubly stochastic matrices, and ωn denote the convex polytope of all n x n doubly substochastic matrices. For a matrix A ϵ ωn, 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 ωn,k denote the subset of ωn which contains all doubly substochastic matrices with sub-defect k. For π a permutation of symmetric group of degree n, the sequence of elements a1π(1); a2π(2), ..., anπ(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 ωn,k, respectively. In this paper, existing results of h and l functions are extended from Ωn to ωn,k. In addition, an analogue of Sylvesters law of the h function on ωn,k is proved.
Cao, Lei; Chen, Zhi; Duan, Xuefeng; Koyuncu, Selcuk; and Li, Huilan, "Diagonal Sums of Doubly Substochastic Matrices" (2019). Mathematics Faculty Articles. 279.