numpy.linalg.cholesky#
- linalg.cholesky(a, /, *, upper=False)[source]#
Cholesky decomposition.
Return the lower or upper Cholesky decomposition,
L * L.H
orU.H * U
, of the square matrixa
, whereL
is lower-triangular,U
is upper-triangular, and.H
is the conjugate transpose operator (which is the ordinary transpose ifa
is real-valued).a
must be Hermitian (symmetric if real-valued) and positive-definite. No checking is performed to verify whethera
is Hermitian or not. In addition, only the lower or upper-triangular and diagonal elements ofa
are used. OnlyL
orU
is actually returned.- Parameters:
- a(…, M, M) array_like
Hermitian (symmetric if all elements are real), positive-definite input matrix.
- upperbool
If
True
, the result must be the upper-triangular Cholesky factor. IfFalse
, the result must be the lower-triangular Cholesky factor. Default:False
.
- Returns:
- L(…, M, M) array_like
Lower or upper-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.
See also
scipy.linalg.cholesky
Similar function in SciPy.
scipy.linalg.cholesky_banded
Cholesky decompose a banded Hermitian positive-definite matrix.
scipy.linalg.cho_factor
Cholesky decomposition of a matrix, to use in
scipy.linalg.cho_solve
.
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
>>> import numpy as np >>> 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 >>> np.linalg.cholesky(np.matrix(A)) matrix([[ 1.+0.j, 0.+0.j], [ 0.+2.j, 1.+0.j]]) >>> # The upper-triangular Cholesky factor can also be obtained. >>> np.linalg.cholesky(A, upper=True) array([[1.-0.j, 0.-2.j], [0.-0.j, 1.-0.j]])