2norm

2norm, usually called the L2 norm or Euclidean norm, is a standard way to measure length in Euclidean space. For a vector x in n-dimensional real or complex space, the 2-norm is defined as ||x||2 = sqrt(x1^2 + x2^2 + ... + xn^2) = sqrt(x^T x). It represents the length of the vector and induces the usual Euclidean distance: the distance between x and y is ||x − y||2. The 2-norm satisfies nonnegativity, definiteness, homogeneity, and the triangle inequality, with ||αx||2 = |α| ||x||2 and ||x + y||2 ≤ ||x||2 + ||y||2. It relates to the inner product by ||x||2^2 = ⟨x, x⟩, and is subject to the Cauchy–Schwarz inequality |⟨x, y⟩| ≤ ||x||2 ||y||2.

For matrices, the term 2-norm often refers to the induced or operator norm: ||A||2 = max_{x ≠ 0} ||Ax||2 / ||x||2. This equals the largest singular value σmax(A) of A and, equivalently, sqrt(λmax(A^*A)) for complex A (or A^T A for real A). If A is real symmetric positive semidefinite, ||A||2 equals the largest eigenvalue in magnitude. The matrix 2-norm is submultiplicative: ||AB||2 ≤ ||A||2 ||B||2, and the condition number is κ2(A) = ||A||2 ||A^{-1}||2 = σmax(A)/σmin(A) when A is invertible. Computation is typically achieved via singular value decomposition A = U Σ V*, with ||A||2 = σmax(A).

In applications, the 2-norm is favored for its geometric interpretation and its compatibility with the dot product, making it central in numerical analysis, optimization, and data analysis. It is one member of the family of p-norms, with p = 2 offering a balance between sensitivity to large components and mathematical convenience.