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Norm (mathematics) facts for kids

Kids Encyclopedia Facts

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 p with the following three properties:

  1. Scales for real numbers a, that is, p(ax) = |a|p(x).
  2. Function of sum is less than sum of functions, that is, p(x + y) \leq p(x) + p(y) (also known as the triangle inequality).
  3. p(x) = 0 if and only if x = 0.

Definition

For a vector x, the associated norm is written as ||x||_p, or Lp where p is some value. The value of the norm of x with some length N is as follows:

||x||_p = \sqrt[p]{x_1^p+x_2^p+...+x_N^p}

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

Examples

  1. The one-norm is the sum of absolute values: \|x\|_1 = |x_1| + |x_2| + ... + |x_N|. 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: \|x\|_2 = \sqrt{x_1^2 + x_2^2 + ... + x_N^2}
  3. Maximum norm is the maximum absolute value: \|x\|_{\infty} = \max(|x_1|,|x_2|,...,|x_N|)
  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.

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