cholesky decomposition

Topics

• matrix decomposition

This decomposition is specifically for square symmetric matrices where all values are greater than zero (positive definite matrices). For real-valued matrices (which is common in ML), the decomposition is defined as:

$$A=L{L}^T$$

where A is the matrix being decomposed, L is a lower triangular matrix. It can also be expressed using an upper triangular matrix (U) as $A=U^{T}U$.

from numpy import array
from numpy.linalg import cholesky
 
A = array([
[2, 1, 1],
[1, 2, 1],
[1, 1, 2]])
 
L = cholesky(A)
 
print(L)
print("Reconstruct:", np.allclose(A, L @ L.T))

Cholesky decomposition is used for solving linear least squares problems in linear regression, as well as in simulation and optimization methods. It’s nearly twice as efficient as LU decomposition for symmetric matrices.

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