QR decomposition in SciPy

Introduction

QR decomposition breaks a matrix into two simpler matrices. It helps solve equations and understand matrix properties easily.

Solving systems of linear equations quickly.

Finding the least squares solution in data fitting.

Computing eigenvalues or matrix factorizations.

Improving numerical stability in calculations.

Simplifying matrix operations in machine learning.

Syntax

SciPy

from scipy.linalg import qr
Q, R = qr(A)

A is the input matrix you want to decompose.

Q is an orthogonal matrix and R is an upper triangular matrix.

Examples

Basic QR decomposition of a 2x2 matrix.

SciPy

from scipy.linalg import qr
import numpy as np
A = np.array([[1, 2], [3, 4]])
Q, R = qr(A)

Use mode='economic' to get reduced size matrices for efficiency.

SciPy

Q, R = qr(A, mode='economic')

Enable pivoting to improve numerical stability in some cases.

SciPy

Q, R, P = qr(A, pivoting=True)

Sample Program

This program decomposes matrix A into Q and R. Q is orthogonal, R is upper triangular.

SciPy

from scipy.linalg import qr
import numpy as np

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

# Perform QR decomposition
Q, R = qr(A)

# Print results
print('Matrix Q:')
print(Q)
print('\nMatrix R:')
print(R)

Important Notes

Q is orthogonal, so Q.T @ Q equals the identity matrix.

R is upper triangular, meaning all elements below the diagonal are zero.

QR decomposition is useful for solving Ax = b by rewriting it as Rx = Q.T @ b.

Summary

QR decomposition splits a matrix into Q (orthogonal) and R (upper triangular).

It helps solve equations and analyze matrices easily.

Use scipy.linalg.qr to perform QR decomposition in Python.