QR decomposition

Topics

• matrix decomposition

This technique is applicable to $n\times m$ matrices (not limited to square matrices) and decomposes a matrix (A) into an orthogonal matrix (Q) and an upper triangular matrix (R).

$$A=QR$$

The matrix Q has dimensions $m\times m$, and R is an $m\times n$ upper triangular matrix. QR decomposition is frequently used to solve systems of linear equations.

import numpy as np
 
# Example 3×2 matrix (m ≥ n)
A = np.array([[12, -51],
              [ 6, 167],
              [-4,  24]], dtype=float)
 
# Compute QR decomposition
Q, R = np.linalg.qr(A)
 
print("Q (Orthogonal):\n", Q)
print("R (Upper-triangular):\n", R)
 
# Verify: A == Q @ R
print("Reconstruct:", np.allclose(A, Q @ R))

Related

• LU decomposition