Understanding NumPy Broadcasting

Example 1: Assigning a Scalar to Part of an Array

You can assign a scalar value to a slice of a NumPy array, and NumPy broadcasts the scalar to all elements in that slice:

import numpy as np

a = np.array([1, 2, 3, 4, 5, 6, 7, 8, 9])
a[:3] = 5

print(a)

import numpy as np

a = np.array([1, 2, 3, 4, 5, 6, 7, 8, 9])
a[:3] = 5

print(a)

How It Works:

• a[:3]: Selects the first three elements of array a.
• = 5: Assigns the scalar value 5 to each selected element by broadcasting.
• The first three elements are replaced with 5, while the rest of the array remains unchanged.

Output

[5 5 5 4 5 6 7 8 9]

[5 5 5 4 5 6 7 8 9]

Example 2: Adding a 1D Array to a 2D Array

Consider a 2D array representing some data, and a 1D array that you want to add to each row of the 2D array. Thanks to broadcasting, NumPy automatically stretches the 1D array across the rows of the 2D array.

import numpy as np

# 2D array with shape (3, 3)
a = np.array([
  [1, 2, 3],
  [4, 5, 6],
  [7, 8, 9]
])

# 1D array with shape (3,)
b = np.array([10, 20, 30])

# Add b to each row of a using broadcasting
result = a + b

print(result)

import numpy as np

# 2D array with shape (3, 3)
a = np.array([
  [1, 2, 3],
  [4, 5, 6],
  [7, 8, 9]
])

# 1D array with shape (3,)
b = np.array([10, 20, 30])

# Add b to each row of a using broadcasting
result = a + b

print(result)

How It Works:

• a: A 2D array of shape (3, 3).
• b: A 1D array of shape (3,).
• Because b has a size of 3, NumPy broadcasts it across each row of a when performing addition.
• The result is a new array where each row of a has had b added element-wise.

Output

[[11 22 33]
 [14 25 36]
 [17 28 39]]

[[11 22 33]
 [14 25 36]
 [17 28 39]]

Shape Compatibility in NumPy Broadcasting

Broadcasting works by comparing the shapes of two arrays element-wise, starting from the trailing dimensions (rightmost). Two dimensions are compatible when:

• They are equal, or
• One of them is 1
• If neither of the above conditions is true, broadcasting is not possible and NumPy will raise a ValueError.

If these conditions are met for all corresponding dimensions, NumPy can broadcast the arrays to a common shape by virtually replicating the smaller array along the dimension where its size is 1.

Here’s an example to illustrate:

import numpy as np

# Array with shape (3, 1)
a = np.array([
    [1],
    [2],
    [3]
    ])

# Array with shape (1, 4)
b = np.array([[10, 20, 30, 40]])

# Shapes are compatible for broadcasting: (3,1) and (1,4)
result = a + b

print("Result shape:", result.shape)
print(result)

import numpy as np

# Array with shape (3, 1)
a = np.array([
    [1],
    [2],
    [3]
    ])

# Array with shape (1, 4)
b = np.array([[10, 20, 30, 40]])

# Shapes are compatible for broadcasting: (3,1) and (1,4)
result = a + b

print("Result shape:", result.shape)
print(result)

Output

Result shape: (3, 4)
[[11 21 31 41]
 [12 22 32 42]
 [13 23 33 43]]

Result shape: (3, 4)
[[11 21 31 41]
 [12 22 32 42]
 [13 23 33 43]]

💡 Tip: Remember, broadcasting doesn’t actually copy data — it just works as if the smaller array were repeated along the compatible dimensions, which keeps operations memory-efficient.

Let’s look at Incompatible shapes:

# Incompatible shapes
a = np.ones((2, 3))   # shape (2,3)
b = np.array([1, 2])  # shape (2,)

# This will raise an error
try:
    result = a + b
except ValueError as e:
    print("Error:", e)

# Incompatible shapes
a = np.ones((2, 3))   # shape (2,3)
b = np.array([1, 2])  # shape (2,)

# This will raise an error
try:
    result = a + b
except ValueError as e:
    print("Error:", e)

Output

Error: operands could not be broadcast together with shapes (2,3) (2,)

Error: operands could not be broadcast together with shapes (2,3) (2,)

💡 Tip: Always check array shapes before operations to avoid broadcasting errors. Using a.shape and b.shape can help debug issues quickly.

Example 3: Adding a Column Vector to a 2D Array

In this example, we’ll add a column vector (a 2D array with shape (3, 1)) to a 2D array with shape (3, 3). NumPy broadcasts the column vector across each column of the 2D array.

import numpy as np

# 2D array with shape (3, 3)
a = np.array([
  [1, 2, 3],
  [4, 5, 6],
  [7, 8, 9]
])

# Column vector with shape (3, 1)
b = np.array([
  [10],
  [20],
  [30]
])

# Add b to each column of a using broadcasting
result = a + b

print(result)

import numpy as np

# 2D array with shape (3, 3)
a = np.array([
  [1, 2, 3],
  [4, 5, 6],
  [7, 8, 9]
])

# Column vector with shape (3, 1)
b = np.array([
  [10],
  [20],
  [30]
])

# Add b to each column of a using broadcasting
result = a + b

print(result)

How It Works:

• a: A 2D array of shape (3, 3).
• b: A column vector of shape (3, 1).
• NumPy broadcasts b across all columns of a — row by row — and performs element-wise addition.

Output

[[11 12 13]
 [24 25 26]
 [37 38 39]]

[[11 12 13]
 [24 25 26]
 [37 38 39]]

💡 Tip: When the smaller array has shape (n, 1), broadcasting stretches it across columns. This is the opposite of using a shape like (1, n), which stretches across rows.