The Cholesky descomposition  

We will study a direct method for solving linear systems: the Cholesky decomposition. Given a symmetric positive definite matrix A, the aim is to build a lower triangular matrix L which has the following property: the product of L and its transpose is equal to A.




Given a symmetric positive definite matrix , the Cholesky decomposition constructs a lower triangular matrix L which has the following property: . A symmetric matrix is positive definite if, for any vector , the product is positive.

The matrix is sometimes called the « square root » of . The Cholesky decomposition is often used to calculate the inverse matrix and the determinant of (equal to the square of the product of the diagonal elements of ).



But as always it is important to learn doing an example, so here it is

Example

If a matrix is defined as


Then the LU factorization can be shown as:



Factoring out the diagonal elements in:



Performing the Cholesky descomposition yields:



Then, verifying the results



So that is it