What is a Toeplitz matrix?

A Toeplitz matrix is a matrix in which all the elements in descending diagonals from left to right are the same. For example, the matrix A given below is a Toeplitz matrix.

If we look at the matrix A carefully, we will see that all the descending diagonals from left to right contains the same element. For example, the principal diagonal contains the element a. The diaginal above the principal diagonal contains the element b and so on.

Please note that a Toeplitz matrix may not be necessarily a square matrix.

How to create a Toeplitz matrix using Python?

We can use the following Python code to create a Toeplitz matrix.

from scipy.linalg import toeplitz

row = [1, 2, 3, 4]
column = [1, 5, 6, 7]

A = toeplitz(c=column, r=row)
print("The Toeplitz Matrix A: \n", A)

Here, the row and column variables specify the elements in the first row and first column of the Toeplitz matrix. The output of the mentioned program will be:

The Toeplitz Matrix A: 
 [[1 2 3 4]
 [5 1 2 3]
 [6 5 1 2]
 [7 6 5 1]]