# IJPAM: Volume 50, No. 4 (2009)

**A PROOF OF HADAMARD INEQUALITY FOR**

NONSINGULAR OR POSITIVE DEFINITE MATRICES

NONSINGULAR OR POSITIVE DEFINITE MATRICES

Bayesian and Interdisciplinary Research Unit

Indian Statistical Institute

203, B.T. Road, Kolkata, 700 108, INDIA

e-mail: sumitra@isical.ac.in

**Abstract.**Hadamard inequality for nonsingular matrices states the following: if
is an nonsingular matrix, then
This result is equivalent to the following fact about positive definite matrices: if
is an positive definite matrix, then
We provide a simple proof of this later fact that uses in a straightforward manner spectral decomposition of a positive definite matrix and the inequality between arithmetic mean and geometric mean of a set of positive numbers.

**Received: **November 11, 2008

**AMS Subject Classification: **15A45

**Key Words and Phrases: **Hadamard inequality, positive definite matrix

**Source:** International Journal of Pure and Applied Mathematics

**ISSN:** 1311-8080

**Year:** 2009

**Volume:** 50

**Issue:** 4