How to create an orthogonal matrix using Python NumPy?

We can use the following Python code to create an orthogonal matrix using NumPy.

import numpy

n = 3
A = numpy.random.randint(low=1, high=10, size=(n, n))

Q, R = numpy.linalg.qr(A, mode="complete")
print("A: \n", A)
print("The Orthogonal Matrix Q: \n", Q)
I = numpy.matmul(Q, numpy.array(Q).transpose()).round()
print("QQ^T: \n", I)

Here, A is a square matrix of random integers. We are using the numpy.random.randint() function to create this matrix. The generated random integers are within the range [low, high). And the size argument specifies the size of the matrix. Here, size=(n, n) specifies that the created matrix has n rows and n columns.

A = numpy.random.randint(low=1, high=10, size=(n, n))

Now, we are using the numpy.linalg.qr() function to create an orthogonal matrix from A. The return value Q contains the orthogonal matrix.

Q, R = numpy.linalg.qr(A, mode="complete")

We can multiply this Q with its transpose to check whether Q is indeed orthogonal. As Q is an orthogonal matrix, we will get an identity matrix when we multiply Q with QT.

I = numpy.matmul(Q, numpy.array(Q).transpose()).round()
print("QQ^T: \n", I)

The output of the mentioned program will be:

A: 
 [[2 8 8]
 [9 8 9]
 [4 1 7]]
The Orthogonal Matrix Q: 
 [[-0.19900744  0.91897461  0.34041403]
 [-0.89553347 -0.02945431 -0.44401829]
 [-0.39801488 -0.3932151   0.82883415]]
QQ^T: 
 [[1. 0. 0.]
 [0. 1. 0.]
 [0. 0. 1.]]