# NumPy: Evaluate Einstein’s summation convention of two given multidimensional arrays

## NumPy: Linear Algebra Exercise-5 with Solution

Write a NumPy program to evaluate Einstein’s summation convention of two given multidimensional arrays.

Note: In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention that implies summation over a set of indexed terms in a formula, thus achieving notational brevity.

**Sample Solution** :

**Python Code :**

```
import numpy as np
a = np.array([1,2,3])
b = np.array([0,1,0])
print("Original 1-d arrays:")
print(a)
print(b)
result = np.einsum("n,n", a, b)
print("Einstein’s summation convention of the said arrays:")
print(result)
x = np.arange(9).reshape(3, 3)
y = np.arange(3, 12).reshape(3, 3)
print("Original Higher dimension:")
print(x)
print(y)
result = np.einsum("mk,kn", x, y)
print("Einstein’s summation convention of the said arrays:")
print(result)
```

Sample Output:

Original 1-d arrays: [1 2 3] [0 1 0] Einstein’s summation convention of the said arrays: 2 Original Higher dimension: [[0 1 2] [3 4 5] [6 7 8]] [[ 3 4 5] [ 6 7 8] [ 9 10 11]] Einstein’s summation convention of the said arrays: [[ 24 27 30] [ 78 90 102] [132 153 174]]

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**Previous:** Write a NumPy program to compute the determinant of a given square array.

**Next:** Write a NumPy program to compute the inner product of vectors for 1-D arrays (without complex conjugation) and in higher dimension.

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## Python: Tips of the Day

**Getting the last element of a list:**

some_list[-1] is the shortest and most Pythonic.

In fact, you can do much more with this syntax. The some_list[-n] syntax gets the nth-to-last element. So some_list[-1] gets the last element, some_list[-2] gets the second to last, etc, all the way down to some_list[-len(some_list)], which gives you the first element.

You can also set list elements in this way. For instance:

>>> some_list = [1, 2, 3] >>> some_list[-1] = 5 # Set the last element >>> some_list[-2] = 3 # Set the second to last element >>> some_list [1, 3, 5]

Note that getting a list item by index will raise an IndexError if the expected item doesn't exist. This means that some_list[-1] will raise an exception if some_list is empty, because an empty list can't have a last element.

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