Matrix norm

Remarks

Clauses 1 to 3 say that $\norm{\cdot}$ is a norm on $\mathbb{K}^{m \times n}$; many authors include submultiplicativity in the word "matrix norm". Example: for a vector norm $\abs{\cdot}$ on $\mathbb{K}^n$, the operator norm $\norm{A} = \max_{\abs{x} = 1} \abs{Ax}$ is a submultiplicative matrix norm, since $\abs{ABx} \le \norm{A} \abs{Bx} \le \norm{A}\norm{B}\abs{x}$. Non-example: $\norm{A}_{\max} = \max_{i,j} \abs{a_{ij}}$ is a norm but not submultiplicative. For $A = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$ one has $A^2 = 2A$, so $\norm{A^2}_{\max} = 2 > 1 = \norm{A}_{\max}^2$. All norms on $\mathbb{K}^{m \times n}$ are equivalent, because the space has finite dimension.