How to create a skew symmetric matrix using Python NumPy?

We can use the following Python code to create a skew symmetric matrix in NumPy.

import numpy

n = 3
A = numpy.random.randint(low=1, high=10, size=(n, n))
B = numpy.triu(A, k=0)
C = B.transpose()
D = B - C
print("A: \n", A)
print("B: \n", B)
print("Skew Symmetric Matrix D: \n", D)

Here, we are first creating a square matrix A 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. size=(n, n) specifies that the created matrix will have n rows and n columns.

In our case, the program generates the following matrix as A:

A: 
 [[9 9 4]
 [5 3 5]
 [6 8 9]]

Now, we are using the numpy.triu() function to create another matrix B such that the elements of B above its principal diagonal are the same as the elements of A above its principal diagonal. And as we are using k=0, the diagonal elements are also copied from A to B. And the rest of the elements of B are zeros. In other words, B is an upper triangular matrix created from A.

So, our program returns the following matrix as B:

B: 
 [[9 9 4]
 [0 3 5]
 [0 0 9]]

Now, we are creating the matrix C which is the transpose of the upper triangular matrix B. So, C is a lower triangular matrix.

C = B.transpose()
D = B – C

So, our program returns the following matrix as C:

C: 
 [[9 0 0]
 [9 3 0]
 [4 5 9]]

Please note that B is an upper triangular matrix. And C is a lower triangular matrix which is also the transpose of B. So, if we subtract C from B and create another matrix D, then the diagonal elements of D will be zeros. And the element in the ith row and jth column of D will be the negative of the element in the jth row and ith column of D. In other words, D will be a skew symmetric matrix.

So, our program returns the following matrix D as the output skew symmetric matrix:

A: 
 [[9 9 4]
 [5 3 5]
 [6 8 9]]
B: 
 [[9 9 4]
 [0 3 5]
 [0 0 9]]
C: 
 [[9 0 0]
 [9 3 0]
 [4 5 9]]
Skew Symmetric Matrix D: 
 [[ 0  9  4]
 [-9  0  5]
 [-4 -5  0]]