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positive definite
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(Definition)
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The definiteness of a matrix is an important property that has use in many areas of mathematics and even 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.
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].
- 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.
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Cross-references: linear equations, Cholesky decomposition, invertible, determinant, positive, eigenvalues, transpose, complex conjugate, non-zero vector, Hermitian matrix, square, Hessian matrix, areas, property, matrix
There are 19 references to this entry.
This is version 7 of positive definite, born on 2002-02-15, modified 2006-08-05.
Object id is 1967, canonical name is PositiveDefinite.
Accessed 41892 times total.
Classification:
| AMS MSC: | 15A48 (Linear and multilinear algebra; matrix theory :: Positive matrices and their generalizations; cones of matrices) |
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Pending Errata and Addenda
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