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  with the following three properties:
 Scales for real numbers , that is, .
 Function of sum is less than sum of functions, that is,  (also known as the triangle inequality).
  if and only if .

 Definition 
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.

 Examples 
 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.
 Euclidean norm (also called L2-norm) is the sum of the squares of the values: 
 Maximum norm is the maximum absolute value: 
When applied to matrices, the Euclidean norm is referred to as the Frobenius norm.
 L0 norm is the number of non-zero elements present in a vector.

 Related pages 

 Magnitude (mathematics)

 References 

Category:Linear algebra
Category:Geometry
Category:Mathematical analysis