Vector outer product with Einstein summation convention in Python

To compute outer product of vectors with Einstein summation convention, use the numpy.einsum() method in Python. The 1st parameter is the subscript. It specifies the subscripts for summation as comma separated list of subscript labels. The 2nd parameter is the operands. These are the arrays for the operation.

The einsum() method evaluates the Einstein summation convention on the operands. Using the Einstein summation convention, many common multi-dimensional, linear algebraic array operations can be represented in a simple fashion. In implicit mode einsum computes these values.

In explicit mode, einsum provides further flexibility to compute other array operations that might not be considered classical Einstein summation operations, by disabling, or forcing summation over specified subscript labels.

Steps

At first, import the required libraries −

import numpy as np

Creating a numpy array using the arange() and reshape() method −

arr = np.arange(5)

Display the array −

print("Our Array...\n",arr)

Check the Dimensions −

print("\nDimensions of our Array...\n",arr.ndim)

Get the Datatype −

print("\nDatatype of our Array object...\n",arr.dtype)

Get the Shape −

print("\nShape of our Array object...\n",arr.shape)

To compute outer product of vectors with Einstein summation convention, use the numpy.einsum() method. The 1st parameter is the subscript. It specifies the subscripts for summation as comma separated list of subscript labels. The 2nd parameter is the operands. These are the arrays for the operation −

print("\nResult (outer product)...\n",np.einsum('i,j', np.arange(2)+1, arr))

Example

import numpy as np

# Creating a numpy array using the arange() and reshape() method
arr = np.arange(5)

# Display the array
print("Our Array...\n",arr)

# Check the Dimensions
print("\nDimensions of our Array...\n",arr.ndim)

# Get the Datatype
print("\nDatatype of our Array object...\n",arr.dtype)

# Get the Shape
print("\nShape of our Array object...\n",arr.shape)

# To compute outer product of vectors with Einstein summation convention, use the numpy.einsum() method in Python.
print("\nResult (outer product)...\n",np.einsum('i,j', np.arange(2)+1, arr))

Output

Our Array...
[0 1 2 3 4]

Dimensions of our Array...
1

Datatype of our Array object...
int64

Shape of our Array object...
(5,)

Result (outer product)...
[[0 1 2 3 4]
[0 2 4 6 8]]