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# Dot

How to find the dot product from two NumPy array.

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### Imports¶

This tutorial imports Plotly and Numpy.

In [1]:
import plotly.plotly as py
import plotly.graph_objs as go

import numpy as np


#### Dot Product¶

To compute the dot product of two NumPy arrays, we can do so with np.dot().

The dot product is a mathematical binary operation which takes, in the computer science case, two arrays of equal length and returns the sum of the pairwise products of the elements in the arrays. To provide clarity to this, if $A$ is an array $[a_1, a_2, ..., a_n]$ and $B$ is an array $[b_1, b_2, ..., b_n]$ both with length $n$, then the dot product is defined as:

\begin{align*} a_1 \times b_1 + a_2 \times b_2 + ... + a_n \times b_n \end{align*}

where $\times$ is just the multiplication operator.

In [2]:
array_a = np.array([2, 3, 0])
array_b = np.array([2, -1, 1])

np.dot(array_a, array_b)

Out[2]:
1

#### Visualize as Vectors¶

We can think of these arrays as defining a vector with each value representing the magnitiude of along the x-axis, y-axis, etc. And the dot product can also be used as a metric for orthogonality between these vectors. In general terms, the larger the number, to more parallel the two vectors are. If the dot product returns $0$, this means the vectors point orthogonally.

We can plot the vectors defined above to get a sense for what the dot product means in that example:

In [3]:
import plotly.plotly as py
import plotly.graph_objs as go

trace1 = go.Scatter3d(
x=[0, array_a[0]],
y=[0, array_a[1]],
z=[0, array_a[2]],
mode='markers+lines',
name='array a',
marker=dict(
size=[12, 1]
)
)

trace2 = go.Scatter3d(
x=[0, array_b[0]],
y=[0, array_b[1]],
z=[0, array_b[2]],
mode='markers+lines',
name='array b',
marker=dict(
size=[12, 1]
)
)

data = [trace1, trace2]
layout = go.Layout(
title = 'Vector Representation of Arrays'
)
fig = go.Figure(data=data, layout=layout)
py.iplot(fig, filename='numpy-min')

Out[3]:
In [2]:
help(np.dot)

Help on built-in function dot in module numpy.core.multiarray:

dot(...)
dot(a, b, out=None)

Dot product of two arrays.

For 2-D arrays it is equivalent to matrix multiplication, and for 1-D
arrays to inner product of vectors (without complex conjugation). For
N dimensions it is a sum product over the last axis of a and
the second-to-last of b::

dot(a, b)[i,j,k,m] = sum(a[i,j,:] * b[k,:,m])

Parameters
----------
a : array_like
First argument.
b : array_like
Second argument.
out : ndarray, optional
Output argument. This must have the exact kind that would be returned
if it was not used. In particular, it must have the right type, must be
C-contiguous, and its dtype must be the dtype that would be returned
for dot(a,b). This is a performance feature. Therefore, if these
conditions are not met, an exception is raised, instead of attempting
to be flexible.

Returns
-------
output : ndarray
Returns the dot product of a and b.  If a and b are both
scalars or both 1-D arrays then a scalar is returned; otherwise
an array is returned.
If out is given, then it is returned.

Raises
------
ValueError
If the last dimension of a is not the same size as
the second-to-last dimension of b.

--------
vdot : Complex-conjugating dot product.
tensordot : Sum products over arbitrary axes.
einsum : Einstein summation convention.
matmul : '@' operator as method with out parameter.

Examples
--------
>>> np.dot(3, 4)
12

Neither argument is complex-conjugated:

>>> np.dot([2j, 3j], [2j, 3j])
(-13+0j)

For 2-D arrays it is the matrix product:

>>> a = [[1, 0], [0, 1]]
>>> b = [[4, 1], [2, 2]]
>>> np.dot(a, b)
array([[4, 1],
[2, 2]])

>>> a = np.arange(3*4*5*6).reshape((3,4,5,6))
>>> b = np.arange(3*4*5*6)[::-1].reshape((5,4,6,3))
>>> np.dot(a, b)[2,3,2,1,2,2]
499128
>>> sum(a[2,3,2,:] * b[1,2,:,2])
499128


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