Numpy.kron(): How to Calculate Kronecker Product Using Numpy?

Let us say we are given a pair of matrices and would like to transform one matrix at the expense of the other. What this means is to scale one matrix using the elements of the other. The mathematical process of making this happen is called the ‘Kronecker Product’ and the NumPy library in Python has an exclusive function that might serve the purpose – kron( ).

This article will help you to understand how to calculate a Kronecker product using the kron( ) function from the numpy library. Let’s get started in understanding this function by first importing the numpy library using the below code.

import numpy as np

Thereafter, we shall explore further the kron( ) function through each of the following sections.

• Syntax of kron( ) function
• Calculating Kronecker Product for Arrays of Same Dimensions
• Calculating Kronecker Product for Arrays of Different Dimensions

Syntax of kron( ) function

It is to be noted that while deploying the kron( ) function, one ought to assign the array that is to be scaled as the second input entity, whilst the array that is to be used for the scaling it as the first input entity.

Following are the inputs that are required for the functioning of the kron( ) function.

numpy.kron(a, b)

where,

• a – N-dimensional array containing the elements to be used for scaling
• b – N-dimensional array which is to be scaled

Calculating Kronecker Product for Arrays of Same Dimensions

In this section let’s find the Kronecker product for a pair of one-dimensional arrays as shown below.

ar1 = np.array([1, 3, 7, 2, 9, 4])
ar2 = np.array([5, 6, 8, 0, -1, 7])

Now let us deploy the kron( ) function for the above two arrays.

np.kron(ar1, ar2)

Once the above code is run, the following computation happens in the back end for calculating the results which shall be returned in the form of a one-dimensional array.

• The first element from ‘ar2’ is multiplied by the first element from ‘ar1’ to deduce the first element of the output array (i.e) 5×1 = 5.
• The second element from ‘ar2’ is multiplied by the first element from ‘ar1’ to deduce the second element of the output array (i.e) 6×1 = 6.
• The same happens with the subsequent elements from ‘ar2’ which gets multiplied by the first element from ‘ar1’ such that 8×1 = 8, 0x1 = 0, -1×1 = -1, 7×1 = 7.
• Now the cycle repeats with the second element from ‘ar1’ such that each element of ‘ar2’ gets multiplied by the second element of ‘ar1’ to obtain further the elements of the output array, such that, 5×3 = 15, 6×3 = 18, …., 7×3 = 21.
• The process iterates again until all the elements in ‘ar2’ are multiplied by those in ‘ar1’.
• The products of these elements are arranged in a sequence starting from the results of the product between elements of ‘ar2’ with the first element from ‘ar1’, followed by the second element from ‘ar2’ & so on.

Putting the above in the lingua of mathematics,

ar1 =      [a₁₁  a₁₂ …..a_1n
                a₂₁ a₂₂ …..a_2n
 a_m1 a_m2 …..a_mn]

ar2 =      [b₁₁  b₁₂ …..b_1p
                b₂₁ b₂₂ …..b_2p
 b_o1 b_o2 …..b_op]

Kronecker product of ar1 & ar2 =              [a₁₁xb₁₁  a₁₂xb₁₂ …..a_1nxb_1n
                                                                           a_m1xb_o1 a_m2xb_o2 …..a_mn xb_op]

Calculating Kronecker Product for Arrays of Different Dimensions:

The kron( ) function also holds well when arrays of dissimilar dimensions are fed in. But, at times it takes the privilege of prepending the smallest of the inputs with ones (1’s), should the need arise.

ar3 = np.array([[1, 3, 7],
                [2, 9, 4]])
ar4 = np.array([[5, 6],
                [8, 0],
                [-1, 7]])
np.kron(ar3, ar4)

Conclusion:

Now that we have reached the end of this article, hope it has elaborated on how to use the kron( ) function from the numpy library to calculate the Kronecker product for a given pair of tensors. Here’s another article that explains the tensordot( ) function from numpy in Python. There are numerous other enjoyable and equally informative articles in AskPython that might be of great help to those who are looking to level up in Python. Whilst you enjoy those, adios!