Читать книгу Numerical Methods in Computational Finance - Daniel J. Duffy - Страница 91

This is another very important class of matrices. Matrices having this property are highly desirable in applications and algorithms. An matrix is positive definite if:

(5.18)

and positive semidefinite if:

(5.19)

Some necessary conditions for positive semidefiniteness are:

 The diagonal elements of A must be positive.

 A is positive definite if and only if all its eigenvalues are positive.

 The element of A having the greatest absolute value must be on the diagonal of A.

 

An example of a positive definite matrix A is:

To prove this let . Then

We can take the square root of a positive definite matrix A, and it is a well-defined function. Furthermore, we can factor a positive definite matrix A into:

(5.20)

where L is a lower triangular matrix having positive values on its diagonal. Equation (5.20) is called the Cholesky decomposition for A.