The **norm of a vector** refers to the length or the magnitude of a vector. There are different ways to calculate the length. The norm of a vector is a non-negative value. In this tutorial, we will learn how to calculate the different types of norms of a vector.

Norm of a vector x is denoted as: ‖** x**‖

The norm of a vector is a measure of its distance from the origin in the vector space.

To calculate the norm, you can either use Numpy or Scipy. Both offer a similar function to calculate the norm.

In this tutorial we will look at two types of norms that are most common in the field of machine learning.

**These are : **

- L
^{1}Norm - L
^{2}Norm

Table of Contents

## How to Calculate the L^{1} Norm of a Vector?

L1 Norm of a vector is also known as the **Manhattan distance** or **Taxicab norm**. The notation for L^{1} norm of a vector x is ‖** x**‖

^{1}.

To calculate the norm, you need to take the **sum of the absolute vector values. **

Let’s take an example to understand this:

a = [1,2,3,4,5]

For the array above, the L^{1} norm is going to be:

1+2+3+4+5 = 15

Let’s take another example:

a = [-1,-2,3,4,5]

The L^{1} norm of this array is :

|-1|+|-2|+3+4+5 = 15

The L^{1} norm for both the vectors is the same as we consider absolute values while computing it.

### Python Implementation of L^{1} norm

Let’s see how can we calculate L^{1} norm of a vector in Python.

### Using Numpy

The Python code for calculating L^{1} norm using Numpy is as follows :

from numpy import array from numpy.linalg import norm arr = array([1, 2, 3, 4, 5]) print(arr) norm_l1 = norm(arr, 1) print(norm_l1)

Output :

[1 2 3 4 5] 15.0

Let’s try calculating it for the array with negative entries in our example above.

from numpy import array from numpy.linalg import norm arr = array([-1, -2, 3, 4, 5]) print(arr) norm_l1 = norm(arr, 1) print(norm_l1)

Output :

[-1 -2 3 4 5] 15.0

### Using Scipy

To calculate L^{1} using Scipy is not very different from the implementation above.

The code for same is:

from numpy import array from scipy.linalg import norm arr = array([-1, -2, 3, 4, 5]) print(arr) norm_l1 = norm(arr, 1) print(norm_l1)

Output :

[-1 -2 3 4 5] 15.0

The code is exactly similar to the Numpy one.

## How to Calculate L^{2} Norm of a Vector?

The notation for the L^{2} norm of a vector x is ‖** x**‖

^{2}.

To calculate the L^{2} norm of a vector, take the square root of the sum of the squared vector values.

Another name for L^{2} norm of a vector is** Euclidean distance. **This is often used for calculating the error in machine learning models.

The Root Mean square error is the Euclidean distance between the actual output of the model and the expected output.

The goal of a machine learning model is to reduce this error.

Let’s consider an example to understand it.

a = [1,2,3,4,5]

The L^{2} norm for the above is :

sqrt(1^2 + 2^2 + 3^2 + 4^2 + 5^2) = 7.416

L^{2} norm is always a positive quantity since we are squaring the values before adding them.

### Python Implementation

The Python implementation is as follows :

from numpy import array from numpy.linalg import norm arr = array([1, 2, 3, 4, 5]) print(arr) norm_l2 = norm(arr) print(norm_l2)

Output :

[1 2 3 4 5] 7.416198487095663

Here we can see that by default the **norm method **returns the L^{2} norm.

## Conclusion

This tutorial was about calculating L^{1} and L^{2} norms in Python. We used Numpy and Scipy to calculate the two norms. Hope you had fun learning with us!

your article contradicts usual mathematical definitions

– standard L1 is defined such as not to be less than zero.

– you name L2 the square of the standard L2 , put units to your vector you’ll see is not the same

see https://en.wikipedia.org/wiki/Norm_(mathematics)

The Wikipedia link says ‘Euclidean distance of a vector from the origin is a norm, called the Euclidean norm, or 2-norm, which may also be defined as the square root of the inner product of a vector with itself.’ That’s what the definition of L2 norm in this article says too. For more information on mathematical definitions of vector norms, refer to this link : http://pages.cs.wisc.edu/~amos/412/lecture-notes/lecture14.pdf