Topics
This method applies to square matrices and decomposes a matrix (A) into a lower triangular matrix (L) and an upper triangular matrix (U).
The LU decomposition is found using an iterative numerical process and can fail for those matrices that cannot be decomposed or decomposed easily. A variation of this decomposition that is numerically more stable to solve in practice is called the LUP decomposition, or the LU decomposition with partial pivoting:
where P is a permutation matrix which acts to permute the rows of A. This attempts to put large entries in the top-left position of A.
import numpy as np
from scipy.linalg import lu
# Example 3×3 matrix
A = np.array([[4, 3, 2],
[3, 2, 1],
[2, 1, 3]], dtype=float)
# Compute LU decomposition
P, L, U = lu(A)
print("P (Permutation):\n", P)
print("L (Lower-triangular):\n", L)
print("U (Upper-triangular):\n", U)
# Verify: P @ A == L @ U
print("Reconstruct:", np.allclose(P @ A, L @ U))LU decomposition is often used for:
- solving systems of linear equations
- calculating determinants
- finding matrix inverses