Hadamard’s inequality

Let $A=(a_{ij})$ with $1\le i,j\le n\in \mathbb{N}$ be a square matrix with complex coefficients. Then the following inequality holds:

$$|det(A)|\le \prod _{i=1}^n{\left(\sum _{j=1}^n{|a_{ij}|}^2\right)}^{\frac{1}{2}}.$$

Moreover, if $A$ is Hermitian and positive semidefinite, the following inequality holds:

$$det(A)\le \prod _{i=1}^n{a}_{ii},$$

with equality if and only if $A$ is a diagonal matrix.

Title	Hadamard’s inequality
Canonical name	HadamardsInequality
Date of creation	2013-03-22 14:32:21
Last modified on	2013-03-22 14:32:21
Owner	mathwizard (128)
Last modified by	mathwizard (128)
Numerical id	9
Author	mathwizard (128)
Entry type	Theorem
Classification	msc 15A45