Matrix Vector multiplication with Einstein summation convention in Python

Matrix Vector multiplication with Einstein summation convention uses the numpy.einsum() method in Python. This method evaluates the Einstein summation convention on operands, allowing many common multi-dimensional linear algebraic operations to be represented in a simple fashion.

Syntax

numpy.einsum(subscripts, operands)

Parameters:

• subscripts ? String specifying the subscripts for summation as comma separated list of subscript labels
• operands ? Arrays for the operation

Understanding Einstein Summation

For matrix-vector multiplication, the notation 'ij,j' means:

• i represents rows of the matrix
• j represents columns of the matrix and elements of the vector
• The repeated index j is summed over

Example

import numpy as np

# Create a 5x5 matrix and a vector of length 5
matrix = np.arange(25).reshape(5, 5)
vector = np.arange(5)

# Display the arrays
print("Matrix:")
print(matrix)
print("\nVector:")
print(vector)

# Check dimensions and shapes
print(f"\nMatrix shape: {matrix.shape}")
print(f"Vector shape: {vector.shape}")

# Matrix-vector multiplication using einsum
result = np.einsum('ij,j', matrix, vector)
print("\nMatrix-Vector multiplication result:")
print(result)

# Verify with traditional dot product
traditional_result = matrix.dot(vector)
print("\nVerification using dot product:")
print(traditional_result)
print(f"\nResults are equal: {np.array_equal(result, traditional_result)}")

Matrix:
[[ 0  1  2  3  4]
 [ 5  6  7  8  9]
 [10 11 12 13 14]
 [15 16 17 18 19]
 [20 21 22 23 24]]

Vector:
[0 1 2 3 4]

Matrix shape: (5, 5)
Vector shape: (5,)

Matrix-Vector multiplication result:
[ 30  80 130 180 230]

Verification using dot product:
[ 30  80 130 180 230]

Results are equal: True

How It Works

The Einstein summation 'ij,j' performs the following calculation ?

import numpy as np

matrix = np.arange(25).reshape(5, 5)
vector = np.arange(5)

print("Step-by-step calculation for first row:")
print(f"Row 0: {matrix[0]} * {vector} = {matrix[0] * vector}")
print(f"Sum: {np.sum(matrix[0] * vector)}")

print(f"\nRow 1: {matrix[1]} * {vector} = {matrix[1] * vector}")
print(f"Sum: {np.sum(matrix[1] * vector)}")

Step-by-step calculation for first row:
Row 0: [0 1 2 3 4] * [0 1 2 3 4] = [ 0  1  4  9 16]
Sum: 30

Row 1: [5 6 7 8 9] * [0 1 2 3 4] = [ 0  6 14 24 36]
Sum: 80

Comparison with Other Methods

Conclusion

The numpy.einsum() method provides an elegant way to perform matrix-vector multiplication using Einstein summation convention. The notation 'ij,j' clearly shows which dimensions are multiplied and summed, making complex linear algebra operations more readable and explicit.