Linear and Multilinear Algebra ( IF 0.9 ) Pub Date : 2020-09-16 , DOI: 10.1080/03081087.2020.1817296 Rafic Nader 1 , Bassam Mourad 2 , Hassan Abbas 2 , Alain Bretto 1
In this note, we consider the problem of characterizing the conditions under which the positive semi-definite pth root of a positive semi-definite doubly stochastic matrix is doubly stochastic. First, we obtain new sufficient conditions for this problem that improve the existing ones for the case p = 2. In addition, if we let denote the set of all positive semi-definite doubly stochastic matrices, and if for any positive integer p, we define is doubly stochastic, then our next result deals with proving that for , the set is not convex but stars convex with respect to which is the matrix whose all entries are equal to 1/n. Next, we identify a large convex set that sits inside . Furthermore, we use the theory of M-matrices to present a method for constructing elements in . In the end, we investigate in depth the way of finding elements in via the use of eigenvalues.
中文翻译:
关于正半正定双随机矩阵的正半正定pth根
在本说明中,我们考虑表征正半正定双随机矩阵的正半正定p th 根是双随机的条件的问题。首先,我们为这个问题获得了新的充分条件,改进了p = 2情况下的现有充分条件 。此外,如果我们让表示所有的集合半正定双随机矩阵,如果对于任何正整数p,我们定义是双重随机的,那么我们的下一个结果就是证明, 集合不是凸的,但星相对于凸哪一个是所有条目都等于 1/ n的矩阵。接下来,我们确定一个位于内部的大凸集. 此外,我们使用M矩阵的理论来提出一种构造元素的方法. 最后,我们深入研究了在通过使用特征值。