positive definite

Introduction

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

1. In optimizing problems, the definiteness of theHessian matrix determines the quality of an extremal value.The full details can be found onthis 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 $x^t$ isthe transpose of $x$.)

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