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 nonzero 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].
References
 1 M. C. Pease, Methods of Matrix Algebra, Academic Press, 1965
 2 C.R. Johnson, Positive definite matrices, American Mathematical Monthly, Vol. 77, Issue 3 (March 1970) 259264.
Title  positive definite 

Canonical name  PositiveDefinite 
Date of creation  20130322 12:20:03 
Last modified on  20130322 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 