Multiply array of vectors with array of matrices; return array of vectors?

问题:

I've got a numpy array of row vectors of shape (n,3) and another numpy array of matrices of shape (n,3,3). I would like to multiply each of the n vectors with the corresponding matrix and return an array of shape (n,3) of the resulting vectors.

By now I've been using a for loop to iterate through the n vectors/matrices and do the multiplication item by item.

I would like to know if there's a more numpy-ish way of doing this. A way without the for loop that might even be faster.

//edit 1:

As requested, here's my loopy code (with n = 10):

    arr_in = np.random.randn(10, 3)     matrices = np.random.randn(10, 3, 3)      for i in range(arr_in.shape[0]): # 10 iterations         arr_out[i] = np.asarray(np.dot(arr_in[i], matrices[i])) 

回答1:

That dot-product is essentially performing reduction along axis=1 of the two input arrays. The dimensions could be represented like so -

arr_in   :     n   3  matrices : n   3   3 

So, one way to solve it would be to "push" the dimensions of arr_in to front by one axis/dimension, thus creating a singleton dimension at axis=2 in a 3D array version of it. Then, sum-reducing the elements along axis = 1 would give us the desired output. Let's show it -

arr_in   : n   [3]   1  matrices : n   [3]   3 

Now, this could be achieved through two ways.

1) With np.einsum -

np.einsum('ij,ijk->ik',arr_in,matrices) 

2) With NumPy broadcasting -

(arr_in[...,None]*matrices).sum(1) 

Runtime test and verify output (for einsum version) -

回答2:

You could use np.einsum. To get v.dot(M) for each vector-matrix pair, use np.einsum("...i,...ij", arr_in, matrices). To get M.dot(v) use np.einsum("...ij,...i", matrices, arr_in)

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