On fractional Hadamard powers of positive definite matrices

Abstract

Let A = ( a ij ) be a real symmetric n × n positive definite matrix with non-negative entries. We show that A α ≡ ( a ij α ) 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.