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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.

The spectrum of an infinite-dimensional doubly stochastic matrix, when considered as a bounded operator on the sequence space ep with 1 ≥ p < ∞, is contained within the closed unit disc D. In our work, we present an infinite doubly stochastic matrix that exhibits the entire closed unit disc as its spectrum. However, we prove that the points eiπr, where r is an irrational real number, cannot serve as eigenvalues for any doubly stochastic matrices, be it finite or infinite in size. On the other hand, we show that every other point within the closed unit disc can indeed be an eigenvalue of an infinite-dimensional doubly stochastic matrix. In fact, we construct a specific example of an infinite doubly stochastic matrix whose point spectrum precisely consists of D∪{eiπr: r ∈ Q}. Additionally, we investigate the behavior of doubly stochastic matrices in the context of the sequence space e∞, highlighting the contrasts with the ep setting for 1 ≥ p < ∞. [ABSTRACT FROM AUTHOR]