A Determinantal Inequality for Positive Definite Matrices
- 1 January 1961
- journal article
- Published by Canadian Mathematical Society in Canadian Mathematical Bulletin
- Vol. 4 (1), 57-62
- https://doi.org/10.4153/cmb-1961-010-9
Abstract
Let H = (Hi, j) (1 ≦ i, j ≦ n) be an nk × nk matrix with complex coefficients, where each Hi, 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 Hi, j = 0 whenever i ≠ j.The case n = 2 of this theorem is contained in [1].Keywords
This publication has 1 reference indexed in Scilit:
- A note on positive definite matricesProceedings of the Glasgow Mathematical Association, 1956