Compute the Outer Product of Two Given Vectors using NumPy in Python

The outer product of two vectors is a matrix where each element [i, j] is the product of the ith element of the first vector and the jth element of the second vector. NumPy provides the outer() function to calculate this efficiently.

Example: In this example, we calculate the outer product of two small 1D arrays.

import numpy as np

x = np.array([1, 2])
y = np.array([3, 4])
res = np.outer(x, y)
print(res)

[[3 4]
 [6 8]]

Explanation: np.outer(x, y) multiplies each element of x with each element of y to form a 2x2 matrix.

Syntax

numpy.outer(a, b, out=None)

Parameters:

• a: First input vector (1D array or flattened).
• b: Second input vector (1D array or flattened).
• out (Optional): array to store the result.

Returns: A 2D array where each element is a[i] * b[j].

Examples

Example 1: In this example, the outer product of two small arrays is calculated.

import numpy as np

a = np.array([6, 2])
b = np.array([2, 5])
res = np.outer(a, b)
print(res)

[[12 30]
 [ 4 10]]

Explanation: np.outer(a, b) multiplies each element of a with each element of b to form a 2x2 matrix.

Example 2: Here, two 2x2 matrices are flattened automatically and the outer product is computed.

import numpy as np

a = np.array([[1, 3], [2, 6]])
b = np.array([[0, 1], [1, 9]])
res = np.outer(a, b)
print(res)

[[ 0  1  1  9]
 [ 0  3  3 27]
 [ 0  2  2 18]
 [ 0  6  6 54]]

Explanation: np.outer(a, b) first flattens both matrices and then computes the outer product, producing a 4x4 matrix.

Example 3: In this example, two 3x3 matrices are flattened and their outer product is calculated.

import numpy as np

a = np.array([[2, 8, 2], [3, 4, 8], [0, 2, 1]])
b = np.array([[2, 1, 1], [0, 1, 0], [2, 3, 0]])
res = np.outer(a, b)
print(res)

[[ 4  2  2  0  2  0  4  6  0]
 [16  8  8  0  8  0 16 24  0]
 [ 4  2  2  0  2  0  4  6  0]
 [ 6  3  3  0  3  0  6  9  0]
 [ 8  4  4  0  4  0  8 12  0]
 [16  8  8  0  8  0 16 24  0]
 [ 0  0  0  0  0  0 ...

Explanation: np.outer(a, b) flattens both 3x3 matrices into 1D arrays and calculates the outer product, resulting in a 9x9 matrix.