Variations in the sub-defect of doubly substochastic matrices

The sub-defect of a doubly stochastic matrix A A, denoted as s d (A) = ⌈ n - sum (A) ⌉ is defined as the minimum number of rows and columns required to be added to transform the doubly substochastic matrix into a doubly stochastic matrix. Here, n n signifies the matrix size, and sum (A) represents the sum of all entries of A A. This article systematically examines the sub-defect characteristics inherited in doubly stochastic matrices, specifically in the context of symmetric, Hankel-symmetric, and centrosymmetric doubly substochastic matrices. Furthermore, we present illustrative examples to elucidate the practical applicability and significance of our approach in comprehending and manipulating the sub-defect of these specialized matrices.