Euclidnorm

Euclidnorm is the Euclidean norm of a vector in Euclidean space, commonly denoted ||x||2 or simply ||x||. It is defined as the square root of the sum of the squares of the components: ||x||2 = sqrt(x1^2 + x2^2 + ... + xn^2). The squared norm, ||x||2^2 = x1^2 + x2^2 + ... + xn^2, is equal to the inner product x · x.

Interpretation and relationships: The Euclidnorm equals the distance from the origin to the point represented by x. For two vectors, the distance is d(x,y) = ||x - y||2. In inner product terms, ||x||2 = sqrt(x · x). Orthogonal transformations preserve the Euclidnorm, since ||Qx||2 = ||x||2 for any orthogonal matrix Q.

Properties: The Euclidnorm is nonnegative and equals zero only for the zero vector (positive definiteness). It is homogeneous: ||αx||2 = |α| ||x||2 for any scalar α, and it satisfies the triangle inequality: ||x + y||2 ≤ ||x||2 + ||y||2. It is differentiable for x ≠ 0, with gradient ∇||x||2 = x / ||x||2; at x = 0 the gradient is not defined, though a subgradient exists.

Computation and related norms: In R^n, compute ||x||2 as the square root of the sum of squares, typically via the dot product ||x||2^2 = x^T x. The Euclidnorm is a special case of the p-norm family with p = 2. Other norms, such as the L1 or L∞ norms, measure length or size in different ways and have different analytical properties.

Applications and context: The Euclidnorm is widely used in optimization, statistics, machine learning, and computer graphics due to its rotational invariance and smoothness (away from zero). It underpins the Euclidean distance and appears in regularization methods like ridge regression, as well as in many geometric and numerical methods.