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Fri 05 January 2018

# @ Python's Matrix Multiplication Operator

Posted by Aly Sivji in Quick Hits

2017 will forever be etched in our memories as the year Python overtook R to become the leading language for Data Science. There are many factors that play into this: Python's simple syntax, the fantastic PyData ecosystem, and of course buy-in from Python's BDFL.

PEP 465 introduced the @ infix operator that is designated to be used for matrix multiplication. The acceptance and implementation of this proposal in Python 3.5 was a signal to the scientific community that Python is taking its role as a numerical computation language very seriously.

I was a Computational Mathematics major in college so matrices are very near and dear to my heart. Shoutout to Professor Jeff Orchard for having us implement matrix algorithms in C++. His Numerical Linear Algebra course was the best class I've ever taken.

In this post, we will explore the @ operator.

In :
import numpy as np

In :
A = np.matrix('3 1; 8 2')
A

Out:
matrix([[3, 1],
[8, 2]])
In :
B = np.matrix('6 1; 7 9')
B

Out:
matrix([[6, 1],
[7, 9]])
In :
A @ B

Out:
matrix([[25, 12],
[62, 26]])

Let's confirm this works

In :
# element at the top left. i.e. (1, 1) aka (0, 0) in python
A[0, 0] * B[0, 0] + A[0, 1] * B[1, 0]

Out:
25
In :
# element at the top right. i.e. (1, 2) aka (0, 1) in python
A[0, 0] * B[0, 1] + A[0, 1] * B[1, 1]

Out:
12
In :
# element at the bottom left. i.e. (2, 1) aka (1, 0) in python
A[1, 0] * B[0, 0] + A[1, 1] * B[1, 0]

Out:
62
In :
# element at the top right. i.e. (2, 2) aka (1, 1) in python
A[1, 0] * B[0, 1] + A[1, 1] * B[1, 1]

Out:
26
In :
# let's put it in matrix form
result = np.matrix([[A[0, 0] * B[0, 0] + A[0, 1] * B[1, 0], A[0, 0] * B[0, 1] + A[0, 1] * B[1, 1]],
[A[1, 0] * B[0, 0] + A[1, 1] * B[1, 0], A[1, 0] * B[0, 1] + A[1, 1] * B[1, 1]]])
result

Out:
matrix([[25, 12],
[62, 26]])
In :
A @ B == result

Out:
matrix([[ True,  True],
[ True,  True]], dtype=bool)

Looks good!

The Python Data Model specifies that the @ operator invokes __matmul__ and __rmatmul__.

We can overload @ by defining custom behavior for each of the special methods above, but this would result in our API not being Pythonic.

To build Pythonic objects, observe how real Python objects behave.