numpy numpy.dot Matrix operations on arrays of vectors

Example

numpy.dot can be used to multiply a list of vectors by a matrix but the orientation of the vectors must be vertical so that a list of eight two component vectors appears like two eight components vectors:

>>> a
array([[ 1.,  2.],
       [ 3.,  1.]])
>>> b
array([[  1.,   2.,   3.,   4.,   5.,   6.,   7.,   8.],
       [  9.,  10.,  11.,  12.,  13.,  14.,  15.,  16.]])
>>> np.dot(a, b)
array([[ 19.,  22.,  25.,  28.,  31.,  34.,  37.,  40.],
       [ 12.,  16.,  20.,  24.,  28.,  32.,  36.,  40.]])

If the list of vectors is laid out with its axes the other way round (which is often the case) then the array needs to be transposed before and then after the dot operation like this:

>>> b
array([[  1.,   9.],
       [  2.,  10.],
       [  3.,  11.],
       [  4.,  12.],
       [  5.,  13.],
       [  6.,  14.],
       [  7.,  15.],
       [  8.,  16.]])
>>> np.dot(a, b.T).T
array([[ 19.,  12.],
       [ 22.,  16.],
       [ 25.,  20.],
       [ 28.,  24.],
       [ 31.,  28.],
       [ 34.,  32.],
       [ 37.,  36.],
       [ 40.,  40.]])

Although the dot function is very fast, sometimes it is better to use einsum. The equivalent of the above would be:

>>> np.einsum('...ij,...j', a, b)

Which is a little slower but allows a list of vertices to be multiplied by a corresponding list of matrices. This would be a very convoluted process using dot:

>>> a
array([[[ 0,  1],
        [ 2,  3]],
       [[ 4,  5],
        [ 6,  7]],
       [[ 8,  9],
        [10, 11]],
       [[12, 13],
        [14, 15]],
       [[16, 17],
        [18, 19]],
       [[20, 21],
        [22, 23]],
       [[24, 25],
        [26, 27]],
       [[28, 29],
        [30, 31]]])
>>> np.einsum('...ij,...j', a, b)
array([[   9.,   29.],
       [  58.,   82.],
       [ 123.,  151.],
       [ 204.,  236.],
       [ 301.,  337.],
       [ 414.,  454.],
       [ 543.,  587.],
       [ 688.,  736.]])

numpy.dot can be used to find the dot product of each vector in a list with a corresponding vector in another list this is quite messy and slow compared with element-wise multiplication and summing along the last axis. Something like this (which requires a much larger array to be calculated but mostly ignored)

>>> np.diag(np.dot(b,b.T))
array([  82.,  104.,  130.,  160.,  194.,  232.,  274.,  320.])

Dot product using element-wise multiplication and summing

>>> (b * b).sum(axis=-1)
array([  82.,  104.,  130.,  160.,  194.,  232.,  274.,  320.])

Using einsum could be achieved with

>>> np.einsum('...j,...j', b, b)
array([  82.,  104.,  130.,  160.,  194.,  232.,  274.,  320.])