Hadamard powers of some positive matrices

Positivity properties of the Hadamard powers of the matrix [ 1 + x i x j ] for distinct positive real numbers x 1 , … , x n and the matrix [ | cos ⁡ ( ( i − j ) π / n ) | ] are studied. In particular, it is shown that the n × n matrix [ ( 1 + x i x j ) 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 .