What is an orthogonal matrix?

An orthogonal matrix is a square matrix of real numbers such that if we take any two columns or any two rows, the two rows or the two columns will be orthonormal. In other words, if we take any two columns and perform a dot product of the column vectors, the dot product will be zero and the column vectors have unit length. The same is true for any two rows of an orthogonal matrix.

So, in one way, we can say that if Q is an orthogonal matrix, then the following holds true:

Here, Q^T is the transpose of the square matrix Q and I is the identity matrix of the same dimension.

So, in other words, for an orthogonal matrix Q, the transpose of the matrix is equal to its inverse.

The matrices given below are some examples of orthogonal matrices.

How to check whether a matrix is an orthogonal matrix using Python?

We can use the following Python code to check whether a matrix is orthogonal.

import numpy

A = numpy.array([[1, 0], [0, -1]])
A_transpose = A.transpose()
A_inverse = numpy.linalg.inv(A)

print("The matrix A: \n", A)
if numpy.allclose(A_transpose, A_inverse):
    print("The matrix A is orthogonal")
else:
    print("The matrix A is not orthogonal")

Here, A is a square matrix of real numbers. We are calculating the transpose and inverse of the matrix using transpose() and inv() functions, respectively. After that, we are using the numpy.allclose() function to check whether these two matrices are equal.

The output of the given program will be:

The matrix A: 
 [[ 1  0]
 [ 0 -1]]
The matrix A is orthogonal