Understanding the rule of numpy.dot()

Question

I am trying to understand the following rule of numpy.dot():

"When a is ND array, b is MD array(where M>=2). The dot product is defindes as the sum product over the last axis of a and the second-to-last axis of b"

What I want to understand is, how the calculation looks in detail for a specific example:

a = np.array([[[2,3,4],[5,6,7],[1,2,3]],[[1,3,4],[7,1,2],[6,2,1]]])

print(a)

    [[[2 3 4]
    [5 6 7]
    [1 2 3]]

    [[1 3 4]
    [7 1 2]
    [6 2 1]]]

b = np.array([[1 , 2, 3],[4, 5 ,6],[7, 8, 9]])

print (b)

b = [[1 2 3]
    [4 5 6]
    [7 8 9]] 

np.dot(a,b) = [[[ 42  51  60]
              [ 78  96 114]
              [ 30  36  42]]

              [[ 41  49  57]
              [ 25  35  45]
              [ 21  30  39]]]

• How to I get the first value "42" of the dot product?
• What is the last axis of a and what is the second-to last axis of b?

I couldn't seem to figure out how to get the first value. I understood the other rules of the numpy.dot() definition, but not this last one.

Answer 1

From the documentation ofnumpy.dot :

If a is an ND array and b is an MD 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])

In your example, a has shape (2,3,3) and the axis are (0,1,2). So the last axis of a is 2. b has shape (3,3) and axis are (0,1). The meaning of second to last axis is penultimate axis. Since b has just 2 axis, the penultimate axis is 0.

Data along last axis of a: [2,3, 4] Data along penultimate axis of b: [1, 4,7]

sum product = sum([2*1,3*4,4*7]) = 42.

Same logic can be applied for all values of the output.