@= and @ are new operators introduced in Python 3.5 performing matrix multiplication. They are meant to clarify the confusion which existed so far with the operator * which was used either for element-wise multiplication or matrix multiplication depending on the convention employed in that particular library/code. As a result, in the future, the operator * is meant to be used for element-wise multiplication only.

As explained in PEP0465, two operators were introduced:

• A new binary operator A @ B, used similarly as A * B
• An in-place version A @= B, used similarly as A *= B

Matrix Multiplication vs Element-wise Multiplication

To quickly highlight the difference, for two matrices:

A = [[1, 2],    B = [[11, 12],
     [3, 4]]         [13, 14]]

• Element-wise multiplication will yield:

  A * B = [[1 * 11,   2 * 12], 
           [3 * 13,   4 * 14]]
  

• Matrix multiplication will yield:

  A @ B  =  [[1 * 11 + 2 * 13,   1 * 12 + 2 * 14],
             [3 * 11 + 4 * 13,   3 * 12 + 4 * 14]]
  

Usage in Numpy

So far, Numpy used the following convention:

• the * operator (and arithmetic operators in general) were defined as element-wise operations on ndarrays and as matrix-multiplication on numpy.matrix type.

• method/function dot was used for matrix multiplication of ndarrays

Introduction of the @ operator makes the code involving matrix multiplications much easier to read. PEP0465 gives us an example:

# Current implementation of matrix multiplications using dot function
S = np.dot((np.dot(H, beta) - r).T,
            np.dot(inv(np.dot(np.dot(H, V), H.T)), np.dot(H, beta) - r))

# Current implementation of matrix multiplications using dot method
S = (H.dot(beta) - r).T.dot(inv(H.dot(V).dot(H.T))).dot(H.dot(beta) - r)

# Using the @ operator instead
S = (H @ beta - r).T @ inv(H @ V @ H.T) @ (H @ beta - r)

Clearly, the last implementation is much easier to read and interpret as an equation.

Example

class Pizza(object):
    def __init__(self):
        self.toppings = []

    def __call__(self, topping):
        # When using '@instance_of_pizza' before a function definition
        # the function gets passed onto 'topping'.
        self.toppings.append(topping())

    def __repr__(self):
        return str(self.toppings)

pizza = Pizza()

@pizza
def cheese():
    return 'cheese'
@pizza
def sauce():
    return 'sauce'

print pizza
# ['cheese', 'sauce']

This shows that the function/method/class you're defining after a decorator is just basically passed on as an argument to the function/method immediately after the @ sign.

First sighting

The microframework Flask introduces decorators from the very beginning in the following format:

from flask import Flask
app = Flask(__name__)

@app.route("/")
def hello():
    return "Hello World!"

This in turn translates to:

rule      = "/"
view_func = hello
# They go as arguments here in 'flask/app.py'
def add_url_rule(self, rule, endpoint=None, view_func=None, **options):
    pass

Realizing this finally allowed me to feel at peace with Flask.