For an $n\times n$ doubly substochastic matrix A, the sub-defect of A is the smallest integer k so that by adding k rows and columns, A becomes a doubly stochastic matrix. In this paper, we introduce the notion of sub-defect for an arbitrary $I\times I$ doubly substochastic matrix $A=[a_{ij}{]}_{i,j\in I}$ by using cardinal numbers and then show that any doubly substochastic matrix can be turned into a doubly stochastic matrix by adding some rows and columns. Then we show that for such a matrix the sub-defect of A, denoted by $\mathrm{s}\mathrm{d}\left(A\right),$ is a cardinal number that is between the $0,1,\dots ,$ and $\mathrm{c}\mathrm{a}\mathrm{r}\mathrm{d}\left(I\right).$ We will characterize all finitely valued sub-defect matrices. Also, the sub-defect of all summable doubly substochastic matrices will be obtained.

为$n\times n$双次随机矩阵A ， A的子缺陷是最小整数k，因此通过添加k行和列，A变为双随机矩阵。在本文中，我们介绍了任意一个子缺陷的概念$我\times 我$双次随机矩阵$\mathrm{一个}=[{\mathrm{一个}}_{\mathrm{一世}j}{]}_{\mathrm{一世},j\in 我}$通过使用基数，然后表明通过添加一些行和列，可以将任何双重亚随机矩阵变成双重随机矩阵。然后我们证明对于这样的矩阵A的子缺陷，表示为$\mathrm{s}\mathrm{d}\left(\mathrm{一个}\right),$是一个基数，介于$0,1,\dots ,$和$\mathrm{C}\mathrm{一个}\mathrm{r}\mathrm{d}\left(我\right).$我们将表征所有有限值的子缺陷矩阵。此外，将获得所有可求和的双重亚随机矩阵的子缺陷。