What is a symmetric matrix?

Let’s say A is a square matrix and the element in the ith row and jth column of A is A_i,j. The square matrix A is a symmetric matrix if and only if the following condition holds true:

Now, we know that, if A^T is the transpose of the matrix A, then:

So, if A is a symmetric matrix and the element in the ith row and jth column of A and A^T are A_i,j and A^T_i,j, respectively, then we can say:

How to check whether a matrix is a symmetric matrix using Python NumPy?

We can use the following Python code to check whether a matrix is a symmetric matrix using NumPy.

import numpy

n = 3
A = numpy.random.randint(low=1, high=10, size=(n, n))
print("A: \n", A)
if numpy.allclose(A, A.transpose()):
    print("A is symmetric")
else:
    print("A is not symmetric")

Firstly, we are creating a square matrix with random integers. We are using the numpy.random.randint() function for that purpose. The generated random integers will be within the range [low, high). And the size argument specifies the size of the created matrix. Here, size=(n,n) specifies that the created matrix will have n rows and n columns.

Now, we are calculating the transpose of the created matrix using the transpose() function. And then, we are checking whether A and the transpose of A are equal element-wise. We are using the numpy.allclose() function for that purpose.

The output of the mentioned program will be:

A: 
 [[7 9 7]
 [1 6 6]
 [7 4 8]]
A is not symmetric