Hadamard powers of some positive matrices

Positivity properties of the Hadamard powers of the matrix [1+xixj] for distinct positive real numbers x1,…,xn and the matrix [|cos⁡((i−j)π/n)|] are studied. In particular, it is shown that the n×n matrix [(1+xixj)r] is positive semidefinite if and only if r is a nonnegative integer or r>n−2, and for every odd integer n≥3 the n×n matrix [|cos⁡((i−j)π/n)|r] is positive semidefinite if and only if r is a nonnegative even integer or r>n−3.