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Evaluate Einstein’s summation convention of two multidimensional NumPy arrays

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In Python, we can use the einsum() function of the NumPy package to compute Einstein’s summation convention of two given multidimensional arrays.

Syntax: numpy.einsum(subscripts, *operands, out=None) 

Parameters: 

subscripts : str

Specifies the subscripts for summation as comma separated list of subscript labels. An implicit (classical Einstein summation) calculation is performed unless the explicit indicator ‘->’ is included as well as subscript labels of the precise output form.

operands : list of array_like          

These are the arrays for the operation.

out : ndarray, optional                 

If provided, the calculation is done into this array.

Returns: The calculation based on the Einstein summation convention.

Example 1: Einstein’s summation convention of two 2X2 matrices

Python3

# Importing library
import numpy as np
  
# Creating two 2X2 matrix
matrix1 = np.array([[1, 2], [0, 2]])
matrix2 = np.array([[0, 1], [3, 4]])
  
print("Original matrix:")
print(matrix1)
print(matrix2)
  
# Output
result = np.einsum("mk,kn", matrix1, matrix2)
  
print("Einstein’s summation convention of the two matrix:")
print(result)

                    

Output:

Original matrix:
[[1 2]
 [0 2]]
[[0 1]
 [3 4]]
Einstein’s summation convention of the two matrix:
[[6 9]
 [6 8]]

Example 2: Einstein’s summation convention of two 3X3 matrices

Python3

# Importing library
import numpy as np
  
# Creating two 3X3 matrix
matrix1 = np.array([[2, 3, 5], [4, 0, 2], [0, 6, 8]])
matrix2 = np.array([[0, 1, 5], [3, 4, 4], [8, 3, 0]])
  
print("Original matrix:")
print(matrix1)
print(matrix2)
  
# Output
result = np.einsum("mk,kn", matrix1, matrix2)
  
print("Einstein’s summation convention of the two matrix:")
print(result)

                    

Output:

Original matrix:
[[2 3 5]
 [4 0 2]
 [0 6 8]]
[[0 1 5]
 [3 4 4]
 [8 3 0]]
Einstein’s summation convention of the two matrix:
[[49 29 22]
 [16 10 20]
 [82 48 24]]

Example 3: Einstein’s summation convention of two 4X4 matrices

Python3

# Importing library
import numpy as np
  
# Creating two 4X4 matrix
matrix1 = np.array([[1, 2, 3, 5], [4, 4, 0, 2], 
                    [0, 1, 6, 8], [0, 5, 6, 9]])
  
matrix2 = np.array([[0, 1, 9, 2], [3, 3, 4, 4], 
                    [1, 8, 3, 0], [5, 2, 1, 6]])
  
print("Original matrix:")
print(matrix1)
print(matrix2)
  
# Output
result = np.einsum("mk,kn", matrix1, matrix2)
  
print("Einstein’s summation convention of the two matrix:")
print(result)

                    

Output:

Original matrix:
[[1 2 3 5]
 [4 4 0 2]
 [0 1 6 8]
 [0 5 6 9]]
[[0 1 9 2]
 [3 3 4 4]
 [1 8 3 0]
 [5 2 1 6]]
Einstein’s summation convention of the two matrix:
[[34 41 31 40]
 [22 20 54 36]
 [49 67 30 52]
 [66 81 47 74]]


Last Updated : 29 Aug, 2020
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