A Determinantal Inequality for Positive Definite Matrices

Abstract

Let H = (H_{i, j}) (1 ≦ i, j ≦ n) be an nk × nk matrix with complex coefficients, where each H_{i, j} is itself a k × k matrix (n, k ≧ 2). Let |H| denote the determinant of H and let ∥H∥ = |(|H _{i, j}|)| (1 ≦ i, j ≦ n ). The purpose of this note is to prove the following theorem.Theorem. If H is positive definite Hermitian then |H| ≦∥H∥. Moreover, |H| = ∥H∥ if and only if H_{i, j} = 0 whenever i ≠ j.The case n = 2 of this theorem is contained in [1].