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.

Title Hadamard’s inequality HadamardsInequality 2013-03-22 14:32:21 2013-03-22 14:32:21 mathwizard (128) mathwizard (128) 9 mathwizard (128) Theorem msc 15A45