HadamardsInequality 2004-08-08 10:10:57 2006-08-21 03:09:50 Theorem % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Let $A=(a_{ij})$ with $1\leq i,j\leq n\in\mathbb{N}$ be a square matrix with complex coefficients. Then the following inequality holds: $$|\det(A)|\leq \prod_{i=1}^n\left(\sum_{j=1}^n|a_{ij}|^2\right)^\frac{1}{2}.$$ Moreover, if $A$ is Hermitian and positive semidefinite, the following inequality holds: $$\det(A)\leq \prod_{i=1}^n a_{ii},$$ with equality if and only if $A$ is a diagonal matrix.