dot_outer.Rmd

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

Our standard imports to start:

```{python}
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](https://en.wikipedia.org/wiki/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:

```{python}
a = np.arange(5)
b = np.arange(10, 15)
np.dot(a, b)
```

```{python}
# 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:

```{python}
a.dot(b)
```

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

```{python}
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 $Z_{i,j} =
\vec{X_{i,:}} \cdot \vec{Y_{:, j}}$.

```{python}
X = np.array([[0, 1, 2], [3, 4, 5]])
X
```

```{python}
Y = np.array([[7, 8], [9, 10], [11, 12]])
Y
```

```{python}
X @ Y
```

```{python}
X[0, :] @ Y[:, 0]
```

```{python}
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.

```{python}
row_vector = np.array([[1, 3, 2]])
print(row_vector.shape)
row_vector
```

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

```{python}
col_vector = np.array([[2], [0], [1]])
print(col_vector.shape)
col_vector
```

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

```{python}
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:

```{python}
col_vector @ row_vector
```

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

```{python}
print(col_vector[0] * row_vector)
print(col_vector[1] * row_vector)
print(col_vector[2] * row_vector)
```

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:

```{python}
np.outer(col_vector.ravel(), row_vector.ravel())
```

## 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,):

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

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:

```{python}
# 1D vector is row vector on the left hand side of dot / matrix multiply.
v @ X
```

```{python}
# 1D vector is column vector on the right hand side of dot / matrix multiply.
w = np.array([-1, 0, 1])
X @ w
```

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:

```{python}
# 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
```