Variations in the sub-defect of doubly substochastic matrices

Abstract

The sub-defect of a doubly stochastic matrix AA, denoted as sd(A)=⌈n−sum(A)⌉sd\left(A)=\lceil n-{\rm{sum}}\left(A)\rceil , 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, nn signifies the matrix size, and sum(A){\rm{sum}}\left(A) represents the sum of all entries of AA. 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.

Keywords

• doubly substochastic matrices
• doubly stochastic matrices
• sub-defect
• permutation matrices
• hankel matrices and centrosymmetric matrices
• 05a05
• 15a03
• 15b99
• 52a20