Introduction
This challenge introduces you to numpy.einsum, a powerful and versatile function for tensor operations. Instead of calling specific functions like dot, trace, or multiply, einsum allows you to define these operations and more using a simple string-based notation. By completing this challenge, you will gain a solid understanding of how to use einsum for common and complex array manipulations.
Task 1: Matrix Multiplication
Matrix multiplication is a fundamental operation in linear algebra. With einsum, you can express this operation by defining how the indices of the input arrays should be combined. For two matrices $A$ (shape $m \times n$) and $B$ (shape $n \times p$), the product $C$ (shape $m \times p$) is defined. In einsum notation, this is ij,jk->ik. The repeated index j is summed over.
Your Task
Complete the matmul function in the file matmul.py. This function should use numpy.einsum to multiply two matrices, A and B, and return the result.
File to Edit
/home/labex/project/matmul.py
The file has been created for you with the following content:
import numpy as np
def matmul(A: np.ndarray, B: np.ndarray) -> np.ndarray:
## TODO: Perform matrix multiplication using Numpy's einsum function.
pass
Task 2: Trace of a Matrix
The trace of a square matrix is the sum of the elements on its main diagonal. einsum provides a very concise way to specify this summation. For a matrix $A$, the diagonal elements are those where the row index equals the column index ($A_{ii}$). The einsum string for this operation is ii->. The repeated index i indicates summation, and the empty output means the result is a scalar.
Your Task
Complete the trace function in trace_of_matrix.py. Use numpy.einsum to calculate the trace of a given square matrix A.
File to Edit
/home/labex/project/trace_of_matrix.py
The file has been created for you with the following content:
import numpy as np
def trace(A: np.ndarray) -> float:
## TODO: Compute the trace of a matrix using Numpy's einsum function.
pass
Task 3: Hadamard Product
The Hadamard product, or element-wise product, creates a new matrix where each element is the product of the corresponding elements from two input matrices of the same shape. For two matrices $A$ and $B$ of shape $(m \times n)$, the Hadamard product $C$ is also of shape $(m \times n)$, where $C_{ij} = A_{ij} \times B_{ij}$. The einsum string for this is ij,ij->ij.
Your Task
Complete the hadamard_product function in hadamard_product.py. This function should compute the element-wise product of two matrices, A and B, using numpy.einsum.
File to Edit
/home/labex/project/hadamard_product.py
The file has been created for you with the following content:
import numpy as np
def hadamard_product(A: np.ndarray, B: np.ndarray) -> np.ndarray:
## TODO: Compute the Hadamard product of two matrices using Numpy's einsum function.
pass
Task 4: Tensor Contraction
einsum truly shines when dealing with higher-dimensional arrays, or tensors. Tensor contraction is a generalization of matrix multiplication. In this task, you will contract a 3D tensor $A$ of shape $(m \times n \times p)$ with a 2D matrix $B$ of shape $(p \times q)$. The goal is to sum over the shared dimension p, resulting in a new tensor of shape $(m \times n \times q)$. The einsum string for this operation is ijk,kl->ijl.
Your Task
Complete the tensor_contract function in tensor_contract.py. Use numpy.einsum to perform a contraction between a 3D tensor A and a 2D matrix B.
File to Edit
/home/labex/project/tensor_contract.py
The file has been created for you with the following content:
import numpy as np
def tensor_contract(A: np.ndarray, B: np.ndarray) -> np.ndarray:
## TODO: Perform tensor contraction between two tensors using Numpy's einsum function.
pass
Summary
Congratulations on completing the NumPy Einsum Challenge! You have successfully used einsum to perform matrix multiplication, calculate a matrix trace, compute the Hadamard product, and perform tensor contraction. This demonstrates your ability to use Einstein summation notation to write concise, efficient, and readable code for complex array operations.
