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In this note, we consider the problem of characterizing the conditions under which the positive semi-definite  p th 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 letdenote the set of allpositive semi-definite doubly stochastic matrices, and if for any positive integer  p , we defineis doubly stochastic, then our next result deals with proving that for, the setis not convex but stars convex with respect towhich is thematrix 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 invia the use of eigenvalues.

On the positive semi-definite p th roots of positive semi-definite doubly stochastic matrices