What is the multiplication operator actually doing with numpy arrays? [duplicate]

Question

I am learning NumPy and I am not really sure what is the operator * actually doing. It seems like some form of multiplication, but I am not sure how is it determined. From ipython:

In [1]: import numpy as np  In [2]: a=np.array([[1,2,3]])  In [3]: b=np.array([[4],[5],[6]])  In [4]: a*b Out[4]:  array([[ 4,  8, 12],        [ 5, 10, 15],        [ 6, 12, 18]])  In [5]: b*a Out[5]:  array([[ 4,  8, 12],        [ 5, 10, 15],        [ 6, 12, 18]])  In [6]: b.dot(a) Out[6]:  array([[ 4,  8, 12],        [ 5, 10, 15],        [ 6, 12, 18]])  In [7]: a.dot(b) Out[7]: array([[32]]) 

It seems like it is doing matrix multiplication, but only b multiplied by a, not the other way around. What is going on?

Answer 1

It's a little bit complicated and has to do with the concept of broadcasting and the fact that all numpy operations are element wise.

1. a is a 2D array with 1 row and 3 columns and b is a 2D array with 1 column and 3 rows.
2. If you try to multiply them element by element (which is what numpy tries to do if you do a * b because every basic operation except the dot operation is element wise), it must broadcast the arrays so that they match in all their dimensions.
3. Since the first array is 1x3 and the second is 3x1 they can be broadcasted to 3x3 matrix according to the broadcasting rules. They will look like:

a = [[1, 2, 3],      [1, 2, 3],      [1, 2, 3]]  b = [[4, 4, 4],      [5, 5, 5],      [6, 6, 6]] 

And now Numpy can multiply them element by element, giving you the result:

[[ 4,  8, 12],  [ 5, 10, 15],  [ 6, 12, 18]] 

When you are doing a .dot operation it does the standard matrix multiplication. More in docs