Frobenius norm and trace

Frobenius norm

The euclidean norm (dot-product-and-law-of-cosines)

$$|\mathbf{v}|=\sqrt{\sum _{i=1}^n{\mathbf{v}}_i^2}$$

is the most common measure of length of a vector, the Frobenius norm defined as

$$\Vert A{\Vert }_F=\sqrt{\sum _{i,j}{\mathbf{a}}_{ij}^2}$$

is the matrix generalization of the euclidean norm for vectors.

Trace

Beyond the definition formula, another way of calculating it is summing the diagonal—computing the trace—of the dot product between the matrix and its transpose and taking the square root

$$\Vert A{\Vert }_F=\sqrt{\text{trace}(A^{T}A)}$$

This stems from the fact that the diagonal of $A^{T}A$ contains the squared column norms—or the squared row norms in reverse order $A{A}^T$.

Matrix distance

Replacing $A$ with $A-B$ defines the distance between matrices

$$d(A,B)=\Vert A-B{\Vert }_F$$