Use numpy to multiply a matrix across an array of points?

Question

I've got an array which contains a bunch of points (3D vectors, specifically):

pts = np.array([
    [1, 1, 1],
    [2, 2, 2],
    [3, 3, 3],
    [4, 4, 4],
    [5, 5, 5],
])

And I would like to multiply each one of those points by a transformation matrix:

pts[0] = np.dot(transform_matrix, pts[0])
pts[1] = np.dot(transform_matrix, pts[1])
…
pts[n] = np.dot(transform_matrix, pts[n])

How can I do this efficiently?

Answer 1

I find it helps to write the einsum version first-- after you see the indices you can often recognize that there's a simpler version. For example, starting from

>>> pts = np.random.random((5,3))
>>> transform_matrix = np.random.random((3,3))
>>> 
>>> pts_brute = pts.copy()
>>> for i in range(len(pts_brute)):
...         pts_brute[i] = transform_matrix.dot(pts_brute[i])
...     
>>> pts_einsum = np.einsum("ij,kj->ik", pts, transform_matrix)
>>> np.allclose(pts_brute, pts_einsum)
True

you can see this is simply

>>> pts_dot = pts.dot(transform_matrix.T)
>>> np.allclose(pts_brute, pts_dot)
True

Answer 2

Matrix-matrix multiplication can be thought of as "batch-mode" matrix-vector multiplication, where each column in the second matrix is one of the vectors being multiplied by the first, with the result vectors being the columns of the resulting matrix.

Also note that since (AB)^T = B^TA^T, and therefore (by transposing both sides) ((AB)^T)^T = AB = (B^TA^T)^T you can make a similar statement about the rows of the first matrix being batch-(left-)multiplied by the transpose of the second matrix, with the result vectors being the rows of the matrix product.