The error message:
ValueError: operands could not be broadcast together with shapes (m,n) (p,q)
occurs in NumPy when you try to perform element-wise operations on arrays that have incompatible shapes.
1. What Does Broadcasting Mean?
Broadcasting in NumPy allows arrays with different shapes to be automatically expanded so they can be used in arithmetic operations without explicit looping.
Example of valid broadcasting:
import numpy as np
A = np.array([[1, 2, 3], [4, 5, 6]]) # Shape (2,3)
B = np.array([1, 2, 3]) # Shape (1,3)
C = A + B # Works because B is broadcasted to (2,3)
print(C)
2. Causes and Solutions
Cause 1: Arrays Have Incompatible Shapes
If two arrays have mismatched dimensions that cannot be aligned, broadcasting fails.
Incorrect Code:
import numpy as np
A = np.array([[1, 2, 3], [4, 5, 6]]) # Shape (2,3)
B = np.array([[1, 2], [3, 4]]) # Shape (2,2)
C = A + B # Error: Shapes (2,3) and (2,2) are incompatible
Solution: Ensure compatible shapes.
B = np.array([[1, 2, 3], [4, 5, 6]]) # Now B is also (2,3)
C = A + B # No error
Cause 2: Mismatched Dimensions During Matrix Multiplication
Matrix multiplication (np.dot() or @) requires the inner dimensions to match.
Incorrect Code:
A = np.array([[1, 2], [3, 4]]) # Shape (2,2)
B = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]]) # Shape (3,3)
C = np.dot(A, B) # Error: Inner dimensions do not match (2,2) and (3,3)
Solution: Adjust the matrix shapes.
B = np.array([[1, 2], [3, 4]]) # Shape (2,2)
C = np.dot(A, B) # Works fine
Cause 3: Using a 1D Array Where a 2D Array is Expected
Some operations require 2D arrays instead of 1D arrays.
Incorrect Code:
A = np.array([[1, 2], [3, 4]]) # Shape (2,2)
B = np.array([1, 2, 3]) # Shape (3,)
C = A + B # Error: Shapes (2,2) and (3,) are incompatible
Solution: Reshape B to match A properly.
B = np.array([[1, 2], [3, 4]]) # Now B is (2,2)
C = A + B # No error
Cause 4: Different Axes for Operations
NumPy allows broadcasting along specific axes, but only when rules are met.
Incorrect Code:
A = np.ones((2, 3, 4)) # Shape (2,3,4)
B = np.ones((3, 4)) # Shape (3,4)
C = A + B # Error: Can't align (2,3,4) with (3,4)
Solution: Expand B‘s dimensions.
B = B[np.newaxis, :, :] # Change shape to (1,3,4)
C = A + B # Now works