### 15.3.1 Cholesky decomposition

If M is a square symmetric positive definite matrix, the Cholesky decomposition is M=PTP, where P is a lower triangular matrix. The cholesky command finds the matrix P.

- cholesky takes M, a square symmetric positive definite matrix.
- cholesky(M) returns a symbolic or numeric matrix P given by the Cholesky decomposition.

#### Examples

cholesky([[1,1],[1,5]])

⎡ ⎢ ⎣ |
1 | 0
1 | 2
| ⎤ ⎥ ⎦

cholesky([[3,1],[1,4]])

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ |
√ |
3
| 0
√ |
3
3
√ |
33
3
| ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

cholesky([[1,1],[1,4]])

⎡ ⎢ ⎢ ⎢ ⎣ |
1 | 0
1 |
√ |
3
| ⎤ ⎥ ⎥ ⎥ ⎦

##### Remark.

If the matrix argument A is not a symmetric matrix, cholesky(A) does not return an error, bu instead uses the symmetric matrix B of the the quadratic form q corresponding to the (non symmetric) bilinear form of the matrix A.

#### Example

cholesky([[1,-1],[-1,4]])

or:

cholesky([[1,-3],[1,4]])

⎡ ⎢ ⎢ ⎢ ⎣ |
1 | 0
−1 |
√ |
3
| ⎤ ⎥ ⎥ ⎥ ⎦