Calculate the QR decomposition of a given matrix using NumPy

In this article, we will discuss QR decomposition of a matrix. QR factorization of a matrix is the decomposition of a matrix say ‘A’ into ‘A=QR’ where Q is orthogonal and R is an upper-triangular matrix. We can calculate the QR decomposition of a given matrix with the help of numpy.linalg.qr(). 

Syntax : numpy.linalg.qr(a, mode=’reduced’)

Parameters :

• a : matrix(M,N) which needs to be factored.
• mode : it is optional. It can be :

Example 1:

Python3

import numpy as np


# Original matrix
matrix1 = np.array([[1, 2, 3], [3, 4, 5]])
print(matrix1)

# Decomposition of the said matrix
q, r = np.linalg.qr(matrix1)
print('\nQ:\n', q)
print('\nR:\n', r)

Output:

[[1 2 3]
 [3 4 5]]

Q:
 [[-0.31622777 -0.9486833 ]
 [-0.9486833   0.31622777]]

R:
 [[-3.16227766 -4.42718872 -5.69209979]
 [ 0.         -0.63245553 -1.26491106]]

Example 2:

Python3

import numpy as np


# Original matrix
matrix1 = np.array([[1, 0], [2, 4]])
print(matrix1)

# Decomposition of the said matrix
q, r = np.linalg.qr(matrix1)
print('\nQ:\n', q)
print('\nR:\n', r)

Output:

[[1 0]
 [2 4]]

Q:
 [[-0.4472136  -0.89442719]
 [-0.89442719  0.4472136 ]]

R:
 [[-2.23606798 -3.57770876]
 [ 0.          1.78885438]]

Example 3:

Python3

import numpy as np

# Create a numpy array
arr = np.array([[5, 11, -15], [12, 34, -51],
                [-24, -43, 92]], dtype=np.int32)

print(arr)

# Find the QR factor of array
q, r = np.linalg.qr(arr)
print('\nQ:\n', q)
print('\nR:\n', r)

Output:

[[  5  11 -15]
 [ 12  34 -51]
 [-24 -43  92]]

Q:
 [[-0.18318583 -0.08610905  0.97929984]
 [-0.43964598 -0.88381371 -0.15995231]
 [ 0.87929197 -0.45984624  0.12404465]]

R:
 [[-27.29468813 -54.77256208 106.06459346]
 [  0.         -11.22347731   4.06028083]
 [  0.           0.           4.88017756]]