Operator norm - Wikipedia

Abstraction: Norm measuring maximum vector lengthening factor of bounded linear operators

Key points:

• Operator norm of a linear map T is the infimum of all c such that ||Tv|| <= c||v||; equivalently, the supremum of ||Tv|| over unit vectors
• For matrices with Euclidean norm, operator norm equals the largest singular value (square root of largest eigenvalue of A^T A)
• Operator norm is sub-multiplicative: ||ST|| <= ||S||*||T||; convergence in operator norm implies uniform convergence on bounded sets
• Spectral radius is bounded above by operator norm; equality holds for normal matrices (spectral theorem); violated for non-normal matrices (e.g., Jordan blocks)
• Some induced matrix norms are NP-hard to compute; most can be computed in O(n^2) operations except the 2-norm which needs O(n^3) or approximation
• Space of bounded operators on L2 with operator norm topology is not separable; bounded operators on a Hilbert space form a C*-algebra

Connections: Operator Norm · Bounded Operators · Functional Analysis · Spectral Radius

Source: http://en.wikipedia.org/wiki/Operator_norm