LU decomposition. LU decomposition is a fundamental… | by Shlok Kumar | Feb, 2025

Let’s decompose the next matrix A:

A = [[1, 1, 1],
[4, 3, -1],
[3, 5, 3]]

Use row operations to transform A into an higher triangular matrix U. Observe the multipliers to assemble L.

Row Operations:

• Subtract 4 * Row1 from Row2:

R2 → R2 - 4 * R1

• Subtract 3 * Row1 from Row3:

R3 → R3 - 3 * R1

• Subtract -2 * Row2 from Row3:

R3 → R3 - (-2) * R2

After these steps, we get:

U = [[1,  1,   1],
[0, -1,  -5],
[0,  0, -10]]
L = [[1,  0,  0],
[4,  1,  0],
[3, -2,  1]]

Verify that A = L × U:

import numpy as np
L = np.array([[1, 0, 0],
[4, 1, 0],
[3, -2, 1]])
U = np.array([[1, 1, 1],
[0, -1, -5],
[0, 0, -10]])
A_reconstructed = np.dot(L, U)
print("Reconstructed A:")
print(A_reconstructed)

Output:

Reconstructed A:
[[ 1  1  1]
[ 4  3 -1]
[ 3  5  3]]

The decomposition is appropriate!