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 $p$ with the following three properties:

1. Scales for real numbers $a$, that is, $p(ax)=|a|p(x)$.
2. Function of sum is less than sum of functions, that is, $p(x+y)\le p(x)+p(y)$ (also known as the triangle inequality).
3. $p(x)=0$ if and only if $x=0$.

Definition

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

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

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: $\Vert x{\Vert }_1=|x_1|+|x_2|+...+|x_N|.$ 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: $\Vert x{\Vert }_2=\sqrt{x_1^2+x_2^2+...+x_N^2}$
3. Maximum norm is the maximum absolute value: $\Vert x{\Vert }_{\mathrm{\infty }}=\max (|x_1|,|x_2|,...,|x_N|)$
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.

Related pages

• Magnitude (mathematics)