# 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. 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].

## 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) 259-264.
Title positive definite PositiveDefinite 2013-03-22 12:20:03 2013-03-22 12:20:03 matte (1858) matte (1858) 10 matte (1858) Definition msc 15A48 PositiveSemidefinite NegativeDefinite QuadraticForm EuclideanVectorSpace EuclideanVectorSpace2