Frobenius norm - Knowlpedia

For a real or complex finite matrix $A=(a_{ij})$ , its Frobenius norm is

\|A\|_F=\left(\sum_{i,j}|a_{ij}|^2\right)^{1/2}.

It is the Euclidean norm of its entries, with complex absolute values in the complex case.

Gradient convention

For a vector field, $|\nabla u|^2=\sum_{i,j}|\partial_j u_i|^2$ uses this norm on the Jacobian matrix. It differs from the operator norm, although all matrix norms on a fixed finite-dimensional matrix space are equivalent.