Vector Norms

A vector norm, sometimes represented with a double bar as ∥x∥, is a function that assigns a non-negative length or size to a vector x in n-dimensional space. Norms are essential in mathematics and machine learning for measuring vector magnitudes and calculating distances.

A vector norm satisfies three properties:

• Non-negativity: ∣x∣ > 0| if x ≠ 0, and ∣x∣=0 if and only if x = 0.
• Scalar Multiplication: ∣kx∣ = ∣k∣ ⋅ ∣x∣ for any scalar k.
• Triangle Inequality: ∣x + y∣ ≤ ∣x∣ + ∣y∣.

Types of Vector Norms

The vector norm ∣x∣_p, also known as the p-norm, for p = 1, 2,… is defined as:

| \mathbf{x} |_p = \left( \sum_{i=1}^{n} | x_i |^p \right)^{\frac{1}{p}}

This general formula encompasses several specific norms that are commonly used.

Commonly used norms are:

• L¹ Norm
• L² Norm
• L∞ Norm

Let's discuss these in detail.

L¹ Norm

The L¹ norm, also known as the Manhattan norm or Taxicab norm, is a way to measure the "length" or "magnitude" of a vector by summing the absolute values of its components.

Mathematically, for a vector x = [x₁, x₂, . . ., x_n], the L¹ norm ∣x∣₁​ is defined as:

∣x∣₁ = ∣x₁∣ + ∣x₂∣ + ∣x₃∣ + ... + ∣x_n∣

Example: If x = [3, −4, 2], then the L1 norm is:

∣x∣₁ ​= ∣3∣ + ∣−4∣ + ∣2∣ = 3 + 4 + 2 = 9

L² Norm

The L² norm, also known as the Euclidean norm, is a measure of the "length" or "magnitude" of a vector, calculated as the square root of the sum of the squares of its components.

For a vector x = [x₁, x₂, . . ., x_n], the L² norm ∣x∣₂​ is defined as:

| \mathbf{x} |_2 = \sqrt{x_1^2 + x_2^2 + \dots + x_n^2}

Example: If x = [3, −4, 2], then the L2 norm is:

| \mathbf{x} |_2 = \sqrt{3^2 + (-4)^2 + 2^2}
= \sqrt{9 + 16 + 4}
=√29 ≈ 5.39

L∞ norm

The L∞ norm, also known as the Infinity norm or Max norm, measures the "size" of a vector by taking the largest absolute value among its components. Unlike the L¹ and L² norms, which consider the combined contribution of all components, the L∞ norm focuses solely on the component with the maximum magnitude.

For a vector x = [x₁, x₂, . . ., x_n], the L∞ norm ∣x∣_∞​ is defined as:

∣x∣_∞ ​= max​∣x_i​∣ where 1 ≤ i ≤ n

Example: If x = [3, −4, 2], then the L∞ norm is:

∣x∣∞​= max(∣3∣, ∣−4∣, ∣2∣) = 4

Read More,

• Vector Space
• Linear Algebra

Practice Problem Based on Vector Norm

Question 1. Given the vector x = [4, -3, 7, 1], calculate the L1 norm (Manhattan norm) of the vector.
Question 2. Given the vector x = [1, -2, 2], calculate the L2 norm (Euclidean norm) of the vector.
Question 3. For the vector x = [7, −1, −4, 6], calculate the L∞ norm (Infinity norm) of the vector.
Question 4. If the L2 norm (Euclidean norm) of a vector x = [x₁, x₂, x₃] is 10, and the components of the vector are x₁ = 6 and x₂ = 8, find the value of x₃.

Answer:-

1. 15
2. 3
3. 7
4. 0