Norm (mathematics)

Inmathematics,thenormof avectoris itslength.Avectoris amathematicalobjectthat has asize,called themagnitude,and adirection.For thereal numbers,the only norm is theabsolute value.Forspaceswith moredimensions,the norm can be anyfunction$p$with the following three properties:

1. Scalesfor real numbers$a$,that is,$p(ax)=|a|p(x)$.
2. Function of sum is less than sum of functions,that is,$p(x+y)\leq p(x)+p(y)$(also known as thetriangle 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:$\|x\|_{1}=|x_{1}|+|x_{2}|+...+|x_{N}|.$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; seeManhattan Distance.
2. Euclidean norm(also called L2-norm) is the sum of the squares of the values:$\|x\|_{2}={\sqrt {x_{1}^{2}+x_{2}^{2}+...+x_{N}^{2}}}$
3. Maximum normis the maximum absolute value:$\|x\|_{\infty }=\max(|x_{1}|,|x_{2}|,...,|x_{N}|)$
4. When applied tomatrices,the Euclidean norm is referred to as theFrobenius norm.
5. L0 normis the number of non-zero elements present in a vector.

Related pages

• Magnitude (mathematics)