Norm (mathematics)

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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 .

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