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 is an square Hermitian matrix. If, for any non-zero vector , we have that
then a positive definite matrix. (Here , where is the complex conjugate of , and is the transpose of .)
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 |
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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 |