l2 norm of a vector

2 min read 19-03-2025

The L2 norm, also known as the Euclidean norm, is a fundamental concept in linear algebra and numerous applications across science and engineering. It essentially measures the "length" or magnitude of a vector. This article will delve into the definition, calculation, properties, and applications of the L2 norm.

What is the L2 Norm?

The L2 norm of a vector is the square root of the sum of the squares of its components. For a vector v in n-dimensional space (Rⁿ), represented as v = (v₁, v₂, ..., v_n), the L2 norm, denoted as ||v||₂ or simply ||v||, is calculated as follows:

||v||₂ = √(v₁² + v₂² + ... + v_n²)

This formula directly corresponds to the Pythagorean theorem in higher dimensions. It represents the straight-line distance from the origin (0, 0, ..., 0) to the point represented by the vector v.

Calculating the L2 Norm: Examples

Let's illustrate with some examples:

Example 1:

Consider the vector v = (3, 4). Its L2 norm is:

||v|| = √(3² + 4²) = √(9 + 16) = √25 = 5

This matches the length of the hypotenuse of a right-angled triangle with sides of length 3 and 4.

Example 2:

For the vector w = (1, -2, 2), the L2 norm is:

||w|| = √(1² + (-2)² + 2²) = √(1 + 4 + 4) = √9 = 3

Properties of the L2 Norm

The L2 norm possesses several important properties:

Non-negativity: ||v|| ≥ 0 for all vectors v. The norm is zero only if the vector is the zero vector.
Homogeneity: ||αv|| = |α| ||v|| for any scalar α. Scaling the vector scales the norm by the absolute value of the scalar.
Triangle inequality: ||u + v|| ≤ ||u|| + ||v|| for any vectors u and v. The norm of the sum of two vectors is less than or equal to the sum of their norms. This is a crucial property in many mathematical proofs and applications.

Applications of the L2 Norm

The L2 norm finds extensive use in various fields:

Machine Learning: Calculating distances between data points (e.g., in k-nearest neighbors algorithms). It's also crucial in regularization techniques to prevent overfitting.
Computer Vision: Measuring the difference between images or image features.
Signal Processing: Analyzing the magnitude of signals.
Physics: Representing the magnitude of physical quantities like force or velocity.
Data Analysis: Calculating the magnitude of errors or residuals in statistical models.

L2 Norm vs. Other Norms

It's important to note that the L2 norm is just one type of vector norm. Other common norms include:

L1 Norm: The sum of the absolute values of the vector components. Less sensitive to outliers than the L2 norm.
L∞ Norm: The maximum absolute value of the vector components.

The choice of norm depends on the specific application and the desired properties.

Conclusion

The L2 norm is a powerful tool for measuring the magnitude of vectors. Its intuitive geometric interpretation and useful properties make it a cornerstone of numerous applications in mathematics, computer science, and engineering. Understanding the L2 norm is essential for anyone working with vectors and their applications. This article provides a foundational understanding to build upon as you explore more advanced topics.