numpy - Python - Matrix outer product

Question

Given two matrices

A: m * r
B: n * r

I want to generate another matrix C: m * n, with each entry C_ij being a matrix calculated by the outer product of A_i and B_j.

For example,

A: [[1, 2],
    [3, 4]]

B: [[3, 1],
    [1, 2]]

gives

C: [[[3, 1],  [[1 ,2],
     [6, 2]],  [2 ,4]],
     [9, 3],  [[3, 6],
     [12,4]],  [4, 8]]]

I can do it using for loops, like

    for i in range (A.shape(0)):
      for j in range (B.shape(0)):
         C_ij = np.outer(A_i, B_j)

I wonder If there is a vectorised way of doing this calculation to speed it up?

Answer 1

The Einstein notation expresses this problem nicely

In [85]: np.einsum('ac,bd->abcd',A,B)
Out[85]: 
array([[[[ 3,  1],
         [ 6,  2]],

        [[ 1,  2],
         [ 2,  4]]],


       [[[ 9,  3],
         [12,  4]],

        [[ 3,  6],
         [ 4,  8]]]])