### 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

⎤
⎥
⎥
⎥
⎦