Numpy and linear algebra: How to code Axꞏy?

I’ve some difficulty in matching what Numpy expects when performing the dot product and vector representation in linear algebra, in term of shapes.

Let’s say I’ve a matrix and two column vectors represented by Numpy arrays:

import numpy as np

A = np.array([[1,2],
              [3,1],
              [-5,2]])

x = np.array([[0],
              [2]])

y = np.array([[-2],
              [0],
              [3]])

and I want to compute Axꞏy, Ax being a matrix multiplication and ꞏ being the dot product. This doesn’t work:

# a = Axꞏy
a = (A @ x).dot(y)

shapes (3,1) and (3,1) not aligned: 1 (dim 1) != 3 (dim 0)

Ax:

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

is indeed a column vector and the dot product of the two column vectors is the scalar: -8+0+12=4, this is the result I’m expecting.

What is the correct operation or reshaping required, using vectors as they are defined?

>Solution :

You can’t compute a dot product between two (3, 1) arrays, a dot product A @ B is only valid if the shape of A and B are (n, k) and (k, m).

It looks like you want:

(A@x).T @ y

Output: [[4]]

Or as scalar:

((A@x).T @ y).item()

Output: 4

NB. in the context of 2D arrays @ and dot are equivalent.

Other operations that would have been valid given the shapes:

(A@x) @ y.T

array([[-8,  0, 12],
       [-4,  0,  6],
       [-8,  0, 12]])


(A.T*x) @ y

array([[0],
       [4]])