Sub-defect of product of I×I finite sub-defect matrices.

For a non-empty set I, the sub-defect of an $ I\times I $ I × I doubly substochastic matrix $ A=[a_{ij}]_{i,j \in I}, $ A = [ a ij ] i , j ∈ I , denoted by $ \mathrm {sd}(A), $ sd (A) , is the smallest cardinal number α for which there is a set J with $ \mathrm {card}(J) =\alpha, $ card (J) = α , $ I\cap J =\emptyset, $ I ∩ J = ∅ , and there exists a doubly stochastic matrix $ D=[d_{ij}]_{i,j\in I\cup J} $ D = [ d ij ] i , j ∈ I ∪ J which contains A as a sub-matrix. In this paper, we show the set of all finite sub-defect matrices is closed under multiplication. We also show that the inequality $ \max \{\mathrm {sd}(A), \mathrm {sd}(B)\} \leq \mathrm {sd}(AB) \leq \min \{n, \mathrm {sd}(A) + \mathrm {sd}(B)\} $ max { sd (A) , sd (B) } ≤ sd (AB) ≤ min { n , sd (A) + sd (B) } which is obtained by Lei Cao et al. remains valid for all finite sub-defect matrices. [ABSTRACT FROM AUTHOR]