The Cholesky decomposition only exists for matrices that are both symmetric and positive-definite, all eigenvalues strictly positive — a common category in practice, since covariance matrices in statistics and many matrices arising from least-squares or optimization problems satisfy exactly this property. Because it exploits that special structure, computing a Cholesky decomposition is roughly twice as fast as a general-purpose decomposition like LU, which is why it's the preferred method specifically when you know in advance that a matrix qualifies.
1Understanding np.linalg.cholesky()
The Cholesky decomposition only exists for matrices that are both symmetric and positive-definite, all eigenvalues strictly positive — a common category in practice, since covariance matrices in statistics and many matrices arising from least-squares or optimization problems satisfy exactly this property. Because it exploits that special structure, computing a Cholesky decomposition is roughly twice as fast as a general-purpose decomposition like LU, which is why it's the preferred method specifically when you know in advance that a matrix qualifies.
np.linalg.cholesky() raises a LinAlgError if the matrix isn't positive-definite or isn't symmetric — this failure is itself sometimes used as a quick numerical test for positive-definiteness, since it's cheaper than explicitly computing all the eigenvalues.
import numpy as np
A = np.array([[4, 2], [2, 3]])
L = np.linalg.cholesky(A)
print(L)2Practical Example
Here is a real-world application of np.linalg.cholesky() showing how it is used in production NumPy code.
import numpy as np
A = np.array([[4, 2], [2, 3]])
L = np.linalg.cholesky(A)
reconstructed = L @ L.T
print(np.allclose(reconstructed, A))3Best Practices
Follow these guidelines when working with np.linalg.cholesky():
1. Use Cholesky decomposition specifically for known symmetric, positive-definite matrices, like covariance matrices, where it's faster than more general decompositions
2. Handle the LinAlgError from a failed Cholesky attempt as a signal the matrix isn't positive-definite, rather than assuming it will always succeed
3. Use the resulting lower-triangular matrix L for efficient random sampling from a multivariate normal distribution, one of its most common practical applications
Tip: np.linalg.cholesky() raises a LinAlgError if the matrix isn't positive-definite or isn't symmetric — this failure is itself sometimes used as a quick numerical test for positive-definiteness, since it's cheaper than explicitly computing all the eigenvalues.
import numpy as np
A = np.array([[4, 2], [2, 3]])
L = np.linalg.cholesky(A)
print(L)