Linear and Multilinear Algebra ( IF 0.9 ) Pub Date : 2020-06-22 , DOI: 10.1080/03081087.2020.1781754 Ali Bayati Eshkaftaki 1
For an 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 doubly substochastic matrix 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 is a cardinal number that is between the and We will characterize all finitely valued sub-defect matrices. Also, the sub-defect of all summable doubly substochastic matrices will be obtained.
中文翻译:
I × I 双重次随机矩阵的最小完成
为双次随机矩阵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双次随机矩阵通过使用基数,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,表示为是一个基数,介于和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。