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 $K_n$ denote the set of all $n\times n$ positive semi-definite doubly stochastic matrices, and if for any positive integer p, we define $K_n^{1/p}:=\{A\in K_n:A^{1/p}$ is doubly stochastic$\}$, then our next result deals with proving that for $n\ge 3$, the set $K_n^{1/p}$ is not convex but stars convex with respect to $J_n$ which is the $n\times n$ matrix whose all entries are equal to 1/n. Next, we identify a large convex set that sits inside $K_n^{1/p}$. Furthermore, we use the theory of M-matrices to present a method for constructing elements in $K_n^{1/p}$. In the end, we investigate in depth the way of finding elements in $K_n^{1/p}$ via the use of eigenvalues.

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${ķ}_n$表示所有的集合$n\times n$半正定双随机矩阵，如果对于任何正整数p，我们定义${ķ}_n^{1/p}:=\{\mathrm{一个}\in {ķ}_n：{\mathrm{一个}}^{1/p}$是双重随机的$\}$，那么我们的下一个结果就是证明$n\ge 3$, 集合${ķ}_n^{1/p}$不是凸的，但星相对于凸${Ĵ}_n$哪一个是$n\times n$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${ķ}_n^{1/p}$. 此外，我们使用M矩阵的理论来提出一种构造元素的方法${ķ}_n^{1/p}$. 最后，我们深入研究了在${ķ}_n^{1/p}$通过使用特征值。