Norm (mathematics)

length in a vector space

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 with

  1. Scales for real numbers , that is
  2. Function of sum is less than sum of functions, that is or the triangle inequality
  3. if and only if .

DefinitionEdit

For a vector  , the associated norm is written as   or L  where   is some value. The value of the norm of   with some length   is as follows:

 

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

ExamplesEdit

  1. The one-norm is the sum of absolute values:   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 is the sum of the squares of the values:  
  3. Maximum norm is the maximum absolute value:  
  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