On the range of a Hadamard power of a positive semidefinite matrix

For any n × n complex matrix A and any integer k ⩾ 1, we prove that the span of image of the map x ↦ (Ax)(k) is equal to the range of (AA∗)(k), where X∗ and X(k) denote the conjugate transpose and the kth Hadamard power of a matrix X, respectively. This settles a conjecture, due to Gorni and Tutaj-Gasińska, related to the study of the Jacobian Conjecture.