The definiteness of a matrix is an important property that has use in many areas of mathematics and physics. Below are some examples:
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).
One can show that a Hermitian matrix is positive definite if and only if all its eigenvalues are positive . 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 .
- 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.
|Date of creation||2013-03-22 12:20:03|
|Last modified on||2013-03-22 12:20:03|
|Last modified by||matte (1858)|