IJPAM: Volume 50, No. 4 (2009)

A PROOF OF HADAMARD INEQUALITY FOR
NONSINGULAR OR POSITIVE DEFINITE MATRICES

Sumitra Purkayastha
Bayesian and Interdisciplinary Research Unit
Indian Statistical Institute
203, B.T. Road, Kolkata, 700 108, INDIA
e-mail: [email protected]


Abstract.Hadamard inequality for nonsingular matrices states the following: if ${\bf A} = ((a_{ij}))$ is an $n \times n$ nonsingular matrix, then $[\det({\bf A})]^{2} \leq \prod _{i=1}^{n} (\sum _{j=1}^{n}a_{ij}^{2}).$ This result is equivalent to the following fact about positive definite matrices: if ${\bf B} = ((b_{ij}))$ is an $n \times n$ positive definite matrix, then $\det({\bf B}) \leq \prod _{i=1}^{n} b_{ii}.$ 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