Outer product calculation by numpy einsum

Question

I am trying to dive into the einsum notation. This question and answers have helped me a lot.

But now I can't grasp the machinery of the einsum when calculating outer product:

x = np.array([1, 2, 3])
y = np.array([4, 5, 6])
np.einsum('i,j->ij', x, y)

array([[ 4.,  5.,  6.],
        [ 8., 10., 12.],
        [12., 15., 18.]])

That answer gives a following rule:

By repeating the label i in both input arrays, we are telling einsum that these two axes should be multiplied together.

I can't understand how this multiplication happened if we hadn't provided any repeated axis label in np.einsum('i,j->ij', x, y)?

Could you please give a steps that np.einsum took in this example?

Or maybe more broader question how einsum works when no matching axis labels are given?

Answer 1

In the output of np.einsum('i,j->ij', x, y), element [i,j] is simply the product of element i in x and element j in y. In other words, np.einsum('i,j->ij', x, y)[i,j] = x[i]*y[j].

Compare it to np.einsum('i,i->i', x, y) were element i of output is x[i]*y[i]:

np.einsum('i,i->i', x, y)

[ 4 10 18]

And if a label in input is missing in output, it means the output has calculated the sum along the missing labels axis. Here is a simple example:

np.einsum('i,j->i', x, y)

[15 30 45]

Here the label j in input is missing in output, which is equivalent to summation along axis=1 (corresponding to label j):

np.sum(np.einsum('i,j->ij', x, y), axis=1)

[15 30 45]