A matrix norm is a norm defined on the vector space of matrices \(\mathbb{C}^{m\times n}\).
A matrix norm is sub-multiplicative if for every matrix \(A,B\in \mathbb{C}^{n\times n}\)
\[ \lVert AB\rVert\le \lVert A\rVert \lVert B\rVert. \]A matrix norm \(\lVert \cdot\rVert_M\) is called consistent with a vector norm \(\lVert \cdot\rVert_V\) on \(\mathbb{C}^n\) if for every \(A\in \mathbb{C}^{n\times n}\) and \(x\in \mathbb{C}^n\)
\[ \lVert Ax\rVert_V \le \lVert A\rVert_M \lVert x\rVert_V. \]