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

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 1×3 and the second is 3×1 they can be broadcasted to 3×3 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