positive definite

Introduction

The definiteness of a matrix is an important property that has use in many areas of mathematics and physics. Below are some examples:

1. In optimizing problems, the definiteness of the Hessian matrix determines the quality of an extremal value. The full details can be found on this page (http://planetmath.org/RelationsBetweenHessianMatrixAndLocalExtrema).

Definition [1] Suppose $A$ is an $n\times n$ square Hermitian matrix. If, for any non-zero vector $x$, we have that

$$x^{\ast }Ax>0,$$

then $A$ a positive definite matrix. (Here $x^{\ast }={\overline{x}}^t$, where $\overline{x}$ is the complex conjugate of $x$, and $x^t$ is the transpose of $x$.)

One can show that a Hermitian matrix is positive definite if and only if all its eigenvalues are positive [1]. Thus the determinant of a positive definite matrix is positive, and a positive definite matrix is always invertible. The Cholesky decomposition provides an economical method for solving linear equations involving a positive definite matrix. Further conditions and properties for positive definite matrices are given in [2].

Title	positive definite
Canonical name	PositiveDefinite
Date of creation	2013-03-22 12:20:03
Last modified on	2013-03-22 12:20:03
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	10
Author	matte (1858)
Entry type	Definition
Classification	msc 15A48
Related topic	PositiveSemidefinite
Related topic	NegativeDefinite
Related topic	QuadraticForm
Related topic	EuclideanVectorSpace
Related topic	EuclideanVectorSpace2