Further inequalities involving the Khatri-Rao product

Styan [G.P.H. Styan, Hadamard products and multivariate statistical analysis. Linear Algebra and Its Appl. 6 (1973) 217–240] established an inequality involving the Hadamard product using statistical reasoning in the context of multivariate analysis. In this paper, the inequality is extended to involve the Khatri-Rao product in the non-negative definite matrix case and in the non-singular Hermitian matrix case. The equality conditions for these extensions are given. Also established are counterpart inequalities in the positive definite matrix case.