Norm (mathematics)

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In mathematics, the norm of a vector is its length. 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 .

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]

  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