On fractional Hadamard powers of positive definite matrices

AbstractLet A = (aij) be a real symmetric n × n positive definite matrix with non-negative entries. We show that Aα ≡ (aijα) is positive definite for all real α ⩾ n − 2. Moreover, the lower bound is sharp. We give related results for pairs of quadratic forms and discuss partial generalizations to the case in which A is a complex Hermitian matrix