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
- Scales for real numbers , that is
- Function of sum is less than sum of functions, that is or the triangle inequality
- if and only if .
Definition[change | change source]
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[change | change source]
- 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 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