NumPy · Broadcasting

Broadcasting is how NumPy combines arrays of different shapes without loops. The rules:

1. Line the shapes up from the right.
2. Two dimensions are compatible if they are equal or one of them is 1.
3. A size-1 dimension is stretched (virtually, with no copy) to match.

Examples: (3, 4) + (4,) works because the row is added to every row. (3, 1) * (1, 4) gives a (3, 4) outer product. (3, 4) + (3,) fails, because 4 ≠ 3. To fix it, make the vector a column with [:, None].

import numpy as np

X = np.array([[1., 2., 3.], [4., 5., 6.], [7., 8., 9.], [10., 11., 12.]])
centred = X - X.mean(axis=0)          # (4,3) - (3,)
print(centred)

col = np.arange(1, 4)[:, None]        # (3,1)
row = np.arange(1, 5)[None, :]        # (1,4)
print(col * row)                      # (3,4) multiplication table

try:
    X + np.array([1, 2, 3, 4])
except ValueError as e:
    print("ValueError:", e)
print((X + np.array([1, 2, 3, 4])[:, None])[:, 0])

P = np.array([[0, 0], [3, 4], [6, 8]])
D = np.sqrt(((P[:, None, :] - P[None, :, :]) ** 2).sum(-1))   # pairwise distances
print(D)

Output:

[[-4.5 -4.5 -4.5]
 [-1.5 -1.5 -1.5]
 [ 1.5  1.5  1.5]
 [ 4.5  4.5  4.5]]
[[ 1  2  3  4]
 [ 2  4  6  8]
 [ 3  6  9 12]]
ValueError: operands could not be broadcast together with shapes (4,3) (4,) 
[ 2.  6. 10. 14.]
[[ 0.  5. 10.]
 [ 5.  0.  5.]
 [10.  5.  0.]]