Introduction

Vector normalization is an essential operation in many areas of mathematics, science, and engineering. This operation scales the original vector to a unit vector, which maintains the original vector's direction but sets its length to 1. This post will provide an overview of vector normalization, focusing on p-norms.

Applications

Normalization Definition

The normalization of a vector is calculated by dividing each element of the vector by the vector's norm. This process creates a unit vector that has the same direction as the original vector but a length (norm) of 1.

Normalization with p-norm

For a given vector v, the normalized version v' in terms of a p-norm is given by:

Here, \|v\|_p denotes the p-norm of the vector v. The value of p determines the type of normalization. For example, p=2 leads to normalization in terms of Euclidean distance, and p=1 results in normalization in terms of Manhattan distance.

References

• Normalization - Wikipedia
• Normalized Vector - MathWorld