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A Determinantal Inequality for Positive Definite Matrices

Published online by Cambridge University Press:  20 November 2018

R. C. Thompson
R. C. Thompson*
Affiliation:
University of British Columbia
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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].

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 4 , Issue 1 , January 1961 , pp. 57 - 62
Copyright
Copyright © Canadian Mathematical Society 1961

References

1. Everitt, W. N., A note on positive definite matrices, Proc. Glasgow Math. Assoc. 3 (1958), 173-175.Google Scholar
2. Wedderburn, J. H. M., Lectures on Matrices, Amer. Math. Soc. Colloquium Publications XVII (1934), 16-19 and 63.Google Scholar
3. Mirsky, L., An Introduction to Linear Algebra, (Oxford, 1955), 420.Google Scholar