Get the QR factorization of a given NumPy array

Introduction

QR factorization, also known as QR decomposition, is a technique in linear algebra where a matrix is decomposed into the product of an orthogonal matrix Q and an upper triangular matrix R. This factorization is widely used in numerical analysis, solving linear systems, and eigenvalue computations. NumPy provides a straightforward way to compute this decomposition for any given array.

What is QR Factorization?

In QR factorization, a matrix A is expressed as:

A = Q × R

where Q is an orthogonal matrix (meaning Q.T × Q = I) and R is an upper triangular matrix. This decomposition is helpful because it simplifies many matrix computations.

How to Get QR Factorization Using NumPy

NumPy’s linalg module includes the qr() function to perform QR factorization easily. You just need to pass the matrix (NumPy array) to this function, and it returns the matrices Q and R.

Example Code

import numpy as np

# Define a 3x3 matrix
A = np.array([[12, -51, 4],
              [6, 167, -68],
              [-4, 24, -41]])

# Perform QR factorization
Q, R = np.linalg.qr(A)

print("Matrix Q:\n", Q)
print("Matrix R:\n", R)

Output Explanation

The function returns two matrices:

• Q: An orthogonal matrix where columns are orthonormal vectors.
• R: An upper triangular matrix.

Multiplying these two matrices reconstructs the original matrix A.

Applications of QR Factorization

QR factorization is an important tool in many numerical algorithms, including:

• Solving linear systems more efficiently.
• Computing eigenvalues and eigenvectors.
• Performing least squares regression.
• Matrix inversion and rank determination.

Conclusion

Using NumPy to get the QR factorization of a matrix is a simple and powerful way to leverage linear algebra in Python. The np.linalg.qr() function makes it easy to decompose matrices and apply this technique in various scientific and engineering computations.