np.einsum() for efficient computation in NumPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of np.einsum for matrix multiplication

What is the output of the following code using np.einsum to multiply two matrices?

NumPy

import numpy as np
A = np.array([[1, 2], [3, 4]])
B = np.array([[5, 6], [7, 8]])
result = np.einsum('ik,kj->ij', A, B)
print(result)

A

[[17 20]
 [39 46]]

B

[[12 14]
 [30 32]]

C

[[19 22]
 [43 50]]

D

[[5 12]
 [21 32]]

💡 Hint

Think about how matrix multiplication sums over the shared dimension.

The einsum string 'ik,kj->ij' performs standard matrix multiplication by summing over the k index. The result matches the dot product of A and B.

❓ data_output

intermediate

Sum of diagonal elements using np.einsum

What is the output of this code that sums the diagonal elements of a matrix using np.einsum?

NumPy

import numpy as np
M = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
diag_sum = np.einsum('ii', M)
print(diag_sum)

A 45

B 15

C 9

D 6

💡 Hint

The diagonal elements are where the row and column indices are equal.

The einsum string 'ii' sums elements where the two indices are the same, which are the diagonal elements 1, 5, and 9. Their sum is 15.

❓ visualization

advanced

Visualizing outer product with np.einsum

Which option shows the correct 2D array output of the outer product computed by np.einsum?

NumPy

import numpy as np
x = np.array([1, 2, 3])
y = np.array([4, 5])
out = np.einsum('i,j->ij', x, y)
print(out)

A

[[4 5]
 [8 10]
 [12 15]]

B

[[4 8 12]
 [5 10 15]]

C

[[5 4]
 [10 8]
 [15 12]]

D

[[1 2 3]
 [4 5 6]]

💡 Hint

Outer product multiplies each element of x by each element of y.

The einsum string 'i,j->ij' computes the outer product, resulting in a matrix where each row is x[i] times all elements of y.

🧠 Conceptual

advanced

Understanding einsum subscripts for tensor contraction

Given two 3D arrays A and B with shapes (2,3,4) and (3,4,5), which einsum subscript string correctly contracts over the last two axes of A and the first two axes of B to produce a (2,5) array?

A 'ijk,ljk->il'

B 'ijk,klj->il'

C 'ijk,jlk->il'

D 'ijk,jkl->il'

💡 Hint

Look for matching indices in the middle dimensions to sum over.

The subscript 'ijk,jkl->il' sums over j and k, contracting the last two axes of A and first two of B, resulting in shape (2,5).

🔧 Debug

expert

Identify the error in this einsum usage

What error does the following code raise when executed?

NumPy

import numpy as np
A = np.array([[1, 2], [3, 4]])
B = np.array([[5, 6], [7, 8]])
result = np.einsum('ij,jkl->ik', A, B)
print(result)

A ValueError: einstein sum subscripts string contains too many subscripts for operand 1

B TypeError: unsupported operand type(s) for einsum

C No error, prints [[19 22]\n [43 50]]

D SyntaxError: invalid syntax in einsum string

💡 Hint

Check the number of indices in the subscript string vs array dimensions.

The subscript 'ij,jkl->ik' expects B to have 3 dimensions (indices j,k,l), but B has shape (2,2) with only 2 dimensions, so einsum raises a ValueError.