What is the meaning of "sum product" as mentioned in Numpy documentation?

Question

In the NumPy v1.15 Reference Guide, the documentation for numpy.dot uses the concept of "sum product".

Namely, we read the following:

• If a is an N-D array and b is a 1-D array, it is a sum product over the last axis of a and b.
• If a is an N-D array and b is an M-D array (where M>=2), it is a sum product over the last axis of a and the second-to-last axis of b:
  dot(a, b)[i,j,k,m] = sum(a[i,j,:] * b[k,:,m])

What is the definition for this "sum product" concept?
(Couldn't find such a definition on Wikipedia, for example.)

Answer 1

https://en.wikipedia.org/wiki/Matrix_multiplication

That is, the entry c[i,j] of the product is obtained by multiplying 
term-by-term the entries of the ith row of A and the jth column of B, 
and summing these m products. In other words, c[i,j] is the dot product 
of the ith row of A and the jth column of B.

https://en.wikipedia.org/wiki/Dot_product

Algebraically, the dot product is the sum of the products of the 
corresponding entries of the two sequences of numbers.

In early math classes did you learn to take the matrix product, by running one finger across the rows of A and down the columns of B, mulitplying pairs of numbers and summing them? That motion is part of my intuition of how that product is taken.

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For the 1d second argument case, np.dot and np.matmul produce the same thing, but describe the action differently:

• If a is an N-D array and b is a 1-D array, it is a sum product over the last axis of a and b.

• If the second argument is 1-D, it is promoted to a matrix by appending a 1 to its dimensions. After matrix multiplication the appended 1 is removed.

  In [103]: np.dot([[1,2],[3,4]], [1,2]) Out[103]: array([ 5, 11]) In [104]: np.matmul([[1,2],[3,4]], [1,2]) Out[104]: array([ 5, 11])

Appending the dimension to B, does:

In [105]: np.matmul([[1,2],[3,4]], [[1],[2]])
Out[105]: 
array([[ 5],
       [11]])

This last is a (2,2) with (2,1) => (2,1)

Sometimes it is clearer to express the action in einsum terms:

In [107]: np.einsum('ij,j->i', [[1,2],[3,4]], [1,2])
Out[107]: array([ 5, 11])

j, the last axis of both arrays is the one that gets 'summed'.