numpy.linalg.cholesky — NumPy v1.11 Manual (original) (raw)

numpy.linalg.cholesky(a)[source]¶

Cholesky decomposition.

Return the Cholesky decomposition, L * L.H, of the square matrix a, where L is lower-triangular and .H is the conjugate transpose operator (which is the ordinary transpose if a is real-valued). a must be Hermitian (symmetric if real-valued) and positive-definite. Only L is actually returned.

Parameters:	a : (..., M, M) array_like Hermitian (symmetric if all elements are real), positive-definite input matrix.
Returns:	L : (..., M, M) array_like Upper or lower-triangular Cholesky factor of a. Returns a matrix object if a is a matrix object.
Raises:	LinAlgError If the decomposition fails, for example, if a is not positive-definite.

Notes

New in version 1.8.0.

Broadcasting rules apply, see the numpy.linalg documentation for details.

The Cholesky decomposition is often used as a fast way of solving

$A \mathbf{x} = \mathbf{b}$

(when A is both Hermitian/symmetric and positive-definite).

First, we solve for $\mathbf{y}$ in

$L \mathbf{y} = \mathbf{b},$

and then for $\mathbf{x}$ in

$L.H \mathbf{x} = \mathbf{y}.$

Examples

A = np.array([[1,-2j],[2j,5]]) A array([[ 1.+0.j, 0.-2.j], [ 0.+2.j, 5.+0.j]]) L = np.linalg.cholesky(A) L array([[ 1.+0.j, 0.+0.j], [ 0.+2.j, 1.+0.j]]) np.dot(L, L.T.conj()) # verify that L * L.H = A array([[ 1.+0.j, 0.-2.j], [ 0.+2.j, 5.+0.j]]) A = [[1,-2j],[2j,5]] # what happens if A is only array_like? np.linalg.cholesky(A) # an ndarray object is returned array([[ 1.+0.j, 0.+0.j], [ 0.+2.j, 1.+0.j]])

But a matrix object is returned if A is a matrix object
LA.cholesky(np.matrix(A)) matrix([[ 1.+0.j, 0.+0.j], [ 0.+2.j, 1.+0.j]])