Matrix norm - Wikipedia

Abstraction: Taxonomy of matrix norms: operator, entry-wise, and Schatten types

Key points:

• Three main perspectives: operator norms (induced by vector norms, measure stretching), entry-wise norms (treat matrix as flat vector), Schatten norms (apply p-norm to singular values)
• Frobenius norm (p=q=2, or Schatten-2) = sqrt of sum of squared elements = sqrt of sum of squared singular values; sub-multiplicative, rotation-invariant
• Spectral norm = largest singular value = operator norm induced by vector 2-norm; equals spectral radius for Hermitian matrices
• Nuclear norm (Schatten-1, trace norm) = sum of singular values; convex envelope of rank function, widely used in low-rank optimization
• All norms on finite-dimensional space are equivalent (induce same topology)
• Sub-multiplicative norms satisfy ||AB|| <= ||A|| ||B||; important for stability analysis

Connections: Matrix Norm · Frobenius Norm · Spectral Norm · Singular Value Decomposition · Linear Algebra

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