---- > [!definition] Definition. ([[matrix norm]]) > A **matrix norm** is a [[norm|vector norm]] $\|\cdot\|$ on the [[vector space]] $\mathbb{F}^{M \times N}$ of [[matrix|matrices]] that is also submultiplicative: $\|AB\| \leq \|A\| \|B\|$ > for all $A,B \in \mathbb{F}^{M \times N}$. > [!warning] > This (in some contexts unconventional) definition *requires* that a [[norm]] must be submultiplicative to be a matrix norm. So, there are vector norms on $\mathbb{F}^{M \times N}$ which are not matrix norms. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```