---- > [!definition] Definition. ([[unitarily invariant norm of matrices]]) > A [[norm]] $\|\cdot\|_{\text{UI}}$ on a [[vector space]] $\mathbb{F}^{M \times N}$ of matrices (not necessarily a [[matrix norm]], be careful!) is called **unitarily invariant** if we have $\|UAV\|_{\text{UI}}=\|A\|_{\text{UI}}$ > for all $A \in \mathbb{F}^{M \times N}$ and all [[isometric matrix|isometric]] $A,V$ of conformable size. > [!basicexample] > - [[Frobenius norm is unitarily invariant]] > - [[Spectral norm is unitarily invariant]] ---- #### ---- #### 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 > ```