numpy.linalg.cholesky¶
- 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
(when A is both Hermitian/symmetric and positive-definite).
First, we solve for in
and then for in
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]])