Master Python’s @ Operator: Matrix Multiplication Made Simple

In Python, the

@

symbol is the matrix multiplication operator introduced in Python 3.5; it performs matrix‑matrix and matrix‑vector products and is equivalent to NumPy’s

np.dot()

or

np.matmul()

functions.

Basic Usage

Example of multiplying two 2×2 matrices:

<code>import numpy as np
# define two matrices
A = np.array([[1, 2],
              [3, 4]])
B = np.array([[5, 6],
              [7, 8]])
# matrix multiplication
C = A @ B
print(C)</code>

Output:

<code>[[19 22]
 [43 50]]</code>

Difference from Element‑wise Multiplication

Using the

*

operator performs element‑wise (Hadamard) multiplication, not matrix multiplication:

<code>D = A * B
print(D)</code>

Output:

<code>[[ 5 12]
 [21 32]]</code>

Matrix‑Vector Multiplication

The

@

operator also works between a matrix and a vector:

<code># define matrix and vector
A = np.array([[1, 2],
              [3, 4]])
v = np.array([5, 6])
# matrix‑vector multiplication
result = A @ v
print(result)</code>

Output:

<code>[17 39]</code>

Common Errors and Tips

A dimension mismatch raises an error because the left matrix’s column count must equal the right matrix’s row count:

<code>A = np.array([[1, 2],
               [3, 4]])
B = np.array([[5, 6]])   # shape (1,2)
C = A @ B   # ValueError: shapes (2,2) and (1,2) not aligned</code>

Summary

The

@

operator implements true matrix multiplication following linear‑algebra rules, suitable for both matrix‑matrix and matrix‑vector operations, and differs from the element‑wise

*

operator; using it with NumPy provides efficient linear‑algebra computation.