python - Numpy array broadcasting rules

Question

I'm having some trouble understanding the rules for array broadcasting in Numpy.

Obviously, if you perform element-wise multiplication on two arrays of the same dimensions and shape, everything is fine. Also, if you multiply a multi-dimensional array by a scalar it works. This I understand.

But if you have two N-dimensional arrays of different shapes, it's unclear to me exactly what the broadcasting rules are. This documentation/tutorial explains that: In order to broadcast, the size of the trailing axes for both arrays in an operation must either be the same size or one of them must be one.

Okay, so I assume by trailing axis they are referring to the N in a M x N array. So, that means if I attempt to multiply two 2D arrays (matrices) with equal number of columns, it should work? Except it doesn't...

>>> from numpy import *
>>> A = array([[1,2],[3,4]])
>>> B = array([[2,3],[4,6],[6,9],[8,12]])
>>> print(A)
[[1 2]
 [3 4]]
>>> print(B)
[[ 2  3]
 [ 4  6]
 [ 6  9]
 [ 8 12]]
>>> 
>>> A * B
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
ValueError: shape mismatch: objects cannot be broadcast to a single shape

Since both A and B have two columns, I would have thought this would work. So, I'm probably misunderstanding something here about the term "trailing axis", and how it applies to N-dimensional arrays.

Can someone explain why my example doesn't work, and what is meant by "trailing axis"?

Answer 1

Well, the meaning of trailing axes is explained on the linked documentation page. If you have two arrays with different dimensions number, say one 1x2x3 and other 2x3, then you compare only the trailing common dimensions, in this case 2x3. But if both your arrays are two-dimensional, then their corresponding sizes have to be either equal or one of them has to be 1. Dimensions along which the array has size 1 are called singular, and the array can be broadcasted along them.

In your case you have a 2x2 and 4x2 and 4 != 2 and neither 4 or 2 equals 1, so this doesn't work.