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]]
