Vector and matrix dot products, “np.outer”#

Our standard imports to start:

import numpy as np
import matplotlib.pyplot as plt
# Display array values to 6 digits of precision
np.set_printoptions(precision=6, suppress=True)

Vector dot products#

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

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

In code:

a = np.arange(5)
b = np.arange(10, 15), b)
# The same thing as
np.sum(a * b)  # Elementwise multiplication

dot is also a method of the NumPy array object, and using the method can be neater and easier to read:

Better still, the matrix multiplication operator @ implies a dot product between two vectors:

a @ b

Matrix dot products#

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

X = np.array([[0, 1, 2], [3, 4, 5]])
array([[0, 1, 2],
       [3, 4, 5]])
Y = np.array([[7, 8], [9, 10], [11, 12]])
array([[ 7,  8],
       [ 9, 10],
       [11, 12]])
X @ Y
array([[ 31,  34],
       [112, 124]])
X[0, :] @ Y[:, 0]
X[1, :] @ Y[:, 0]

The outer product#

We can use the rules of matrix multiplication for row vectors and column vectors.

A row vector is a 2D vector where the first dimension is length 1.

row_vector = np.array([[1, 3, 2]])
(1, 3)
array([[1, 3, 2]])

A column vector is a 2D vector where the second dimension is length 1.

col_vector = np.array([[2], [0], [1]])
(3, 1)

We know what will happen if we matrix multiply the row vector and the column vector:

row_vector @ col_vector

What happens when we matrix multiply the column vector by the row vector? We know this will work because we are multiplying a 3 by 1 array by a 1 by 3 array, so this should generate a 3 by 3 array:

col_vector @ row_vector
array([[2, 6, 4],
       [0, 0, 0],
       [1, 3, 2]])

This arises from the rules of matrix multiplication, except there is only one row * column pair making up each of the output elements:

print(col_vector[0] * row_vector)
print(col_vector[1] * row_vector)
print(col_vector[2] * row_vector)
[[2 6 4]]
[[0 0 0]]
[[1 3 2]]

This (M by 1) vector matrix multiply with a (1 by N) vector is also called the outer product of two vectors. We can generate the same thing from 1D vectors, by using the numpy np.outer function:

np.outer(col_vector.ravel(), row_vector.ravel())
array([[2, 6, 4],
       [0, 0, 0],
       [1, 3, 2]])

Dot, vectors and matrices#

Unlike MATLAB, Python has one-dimensional vectors. For example, if I slice a column out of a 2D array of shape (M, N), I do not get a column vector, shape (M, 1), I get a 1D vector, shape (M,):

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

Because the 1D vector has lost the idea of being a column rather than a row in a matrix, it is no longer unambiguous what \(v \cdot \mathbf{X}\) means. It could mean a dot product of a row vector shape (1, M) with a matrix shape (M, N), which is valid – or a dot product of a row vector (M, 1) with a matrix shape (M, N), which is not valid.

If you pass a 1D vector into the dot function or method, or use the @ matrix multiplier, NumPy assumes you mean it to be a row vector on the left, and a column vector on the right, which is nearly always what you intended:

# 1D vector is row vector on the left hand side of dot / matrix multiply.
v @ X
array([ 9, 12, 15])
# 1D vector is column vector on the right hand side of dot / matrix multiply.
w = np.array([-1, 0, 1])
X @ w
array([2, 2])

Notice that, in both cases, @ returns a 1D result.

It sometimes helps to make a 1D vector into a 2D row or column vector, to make your intention explicit, and preserve the 2D shape of the output:

# Turn 1D vector into explicit row vector
row_v = np.reshape(v, (1, 2))
# @ now returns a row vector rather than a 1D vector
row_v @ X
array([[ 9, 12, 15]])