# 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 |