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 the following three properties:[1]

  1. Scales for real numbers , that is, .
  2. Function of sum is less than sum of functions, that is, (also known as the triangle inequality).
  3. if and only if .

Definition change

For a vector  , the associated norm is written as  ,[2] or L  where   is some value. The value of the norm of   with some length   is as follows:[3]


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

Examples change

  1. The one-norm is the sum of absolute values:  [2] 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:[3]  
  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.

Related pages change

References change

  1. "Norm - Encyclopedia of Mathematics". Retrieved 2020-08-24.
  2. 2.0 2.1 "Comprehensive List of Algebra Symbols". Math Vault. 2020-03-25. Retrieved 2020-08-24.
  3. 3.0 3.1 Weisstein, Eric W. "Vector Norm". Retrieved 2020-08-24.