Norm (mathematics)

In mathematics, the norm of a vector is its length. A vector is a mathematical object that has a size, called the magnitude, and a direction. For the real numbers, the only norm is the absolute value. For spaces with more dimensions, the norm can be any function [math]\displaystyle{ p }[/math] with the following three properties:

1. Scales for real numbers [math]\displaystyle{ a }[/math], that is, [math]\displaystyle{ p(ax) = |a|p(x) }[/math].
2. Function of sum is less than sum of functions, that is, [math]\displaystyle{ p(x + y) \leq p(x) + p(y) }[/math] (also known as the triangle inequality).
3. [math]\displaystyle{ p(x) = 0 }[/math] if and only if [math]\displaystyle{ x = 0 }[/math].

Definition

For a vector [math]\displaystyle{ x }[/math], the associated norm is written as [math]\displaystyle{ ||x||_p }[/math], or L[math]\displaystyle{ p }[/math] where [math]\displaystyle{ p }[/math] is some value. The value of the norm of [math]\displaystyle{ x }[/math] with some length [math]\displaystyle{ N }[/math] is as follows:

[math]\displaystyle{ ||x||_p = \sqrt[p]{|x_1|^p+|x_2|^p+...+|x_N|^p} }[/math]

The most common usage of this is the Euclidean norm, also called the standard distance formula.

Examples

1. The one-norm is the sum of absolute values: [math]\displaystyle{ \|x\|_1 = |x_1| + |x_2| + ... + |x_N|. }[/math] This is like finding the distance from one place on a grid to another by summing together the distances in all directions the grid goes; see Manhattan Distance.
2. Euclidean norm (also called L2-norm) is the sum of the squares of the values: [math]\displaystyle{ \|x\|_2 = \sqrt{x_1^2 + x_2^2 + ... + x_N^2} }[/math]
3. Maximum norm is the maximum absolute value: [math]\displaystyle{ \|x\|_{\infty} = \max(|x_1|,|x_2|,...,|x_N|) }[/math]
4. When applied to matrices, the Euclidean norm is referred to as the Frobenius norm.
5. L0 norm is the number of non-zero elements present in a vector.

Norm (mathematics) Media

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  representation of a unit Circle using the Chebyshev distance metric: ‖x‖∞

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  Illustrations of unit circles in different norms.

Related pages

• Magnitude (mathematics)