A matrix norm is a norm on the set $M_{m\times n}(R)$ of $m\times n$ matrices over a ring $R$ , and is usually denoted by $||\cdot||$ .

The Frobenius matrix norm $||\cdot||_F$ and the matrix $p$ -norm $||\cdot||_p$ are the most frequently used matrix norms.

A vector norm is a matrix norm for which the ring $R$ is a field.