Dot product between 2D and 3D numpy arrays

Question

I have 2 arrays x and y with shapes (2, 3, 3), respectively, (3, 3). I want to compute the dot product z with shape (2, 3) in the following way:

x = np.array([[[a111, a121, a131], [a211, a221, a231], [a311, a321, a331]],
               [[a112, a122, a132], [a212, a222, a232], [a312, a322, a332]]])
y = np.array([[b11, b12, b13], [b21, b22, b23], [b31, b32, b33]])
z = np.array([[a111*b11+a121*b12+a131*b13, a211*b21+a221*b22+a231*b23, a311*b31+a321*b32+a331*b33], 
[a112*b11+a122*b12+a132*b13, a212*b21+a222*b22+a232*b23, a312*b31+a322*b32+a332*b33]])

Any ideas on how to do this in a vectorized way?

Answer 1

On the sum-reductions shown in the question, it seems the reduction is along the last axis, while keeping the second axis of x aligned with the first axis of y. Because of that requirement of axis-alignment, we can use np.einsum. Thus, one vectorized solution would be -

np.einsum('ijk,jk->ij',x, y)

Sample run -

In [255]: x
Out[255]: 
array([[[5, 1, 7],
        [2, 1, 7],
        [5, 1, 2]],

       [[6, 4, 7],
        [3, 8, 1],
        [1, 7, 7]]])

In [256]: y
Out[256]: 
array([[5, 4, 7],
       [8, 2, 5],
       [2, 3, 3]])

In [260]: np.einsum('ijk,jk->ij',x, y)
Out[260]: 
array([[78, 53, 19],
       [95, 45, 44]])

In [261]: 5*5 + 1*4 + 7*7
Out[261]: 78

In [262]: 2*8 + 1*2 + 7*5
Out[262]: 53