Dot products in numpy

If I have two vectors \(\mathbf{a}\) with elements \(a_0, a_1, ... a_{n-1}\), and \(\mathbf{b}\) with elements \(b_0, b_1, ... b_{n-1}\) then the dot product is defined as:

\[\mathbf{a}\cdot \mathbf{b} = \sum_{i=0}^{n-1} a_ib_i = a_0b_0 + a_1b_1 + \cdots + a_{n-1}b_{n-1}\]

In code:

>>> a = np.arange(5)
>>> b = np.arange(10, 15)
>>> np.dot(a, b)
130

We could do exactly the same calculation using elementwise multiplication followed by a sum:

>>> np.sum(a * b)  # Elementwise multiplication
130

Matrix multiplication operates by taking dot products of the rows of the first array (matrix) with the columns of the second.

Let’s say I have a matrix \(\mathbf{X}\), and \(X_{i,:}\) is row \(i\) in \(\mathbf{X}\). I have a matrix \(\mathbf{Y}\), and \(Y_{:,j}\) is column \(j\) in \(\mathbf{Y}\). The output matrix \(\mathbf{Z} = \mathbf{X} \mathbf{Y}\) has entry \(Z_{i,j} = X_{i,:} \cdot Y_{:, j}\).

>>> X = np.array([[0, 1, 2], [3, 4, 5]])
>>> X
array([[0, 1, 2],
       [3, 4, 5]])

>>> Y = np.array([[7, 8], [9, 10], [11, 12]])
>>> Y
array([[ 7,  8],
       [ 9, 10],
       [11, 12]])

The numpy array dot method does this operation:

>>> X.dot(Y)
array([[ 31,  34],
       [112, 124]])

>>> X[0, :].dot(Y[:, 0])
31

>>> X[1, :].dot(Y[:, 0])
112