---- > [!definition] Definition. ([[Frobenius norm]]) > The **Frobenius [[norm]]** of a $M \times N$ [[matrix]] $A$ over $\mathbb{R}$ or $\mathbb{C}$ is defined as $\|A\|_{F}:=\sqrt{ \sum_{i,j}^{}{ |a_{ij}|^{2} }}=\|\text{vec}(A)\|_{2}=\sqrt{ \text{Tr } A'A}=\sqrt{ \sum_{\lambda _k \in \mathscr{E}(A'A)}^{} \lambda_{k}}=\sqrt{ \sum_{\sigma _\ell \in \mathscr{S}(A)}^{} \sigma_{\ell}^{2}},$ > where > - $\text{vec}$ denotes the [[vectorize operation]] and $\|\cdot\|_{2}$ denotes the [[Euclidean inner product|Euclidean 2-norm]]; > - $\mathscr{E}(A'A)$ denotes the set of [[eigenvalue]]s of $A'A$; > - $\mathscr{S}(A)$ denotes the set of [[singular values]] of $A$. > \ > Notice that $\|\cdot\|_{F}$ is induced by the [[Frobenius inner product]]. > [!note] > In Julia: $\texttt{norm(A,2)}$ or just $\texttt{norm(A)}$. > [!basicproperties] > - Because the [[group-invariant function|Euclidean norm is unitarily invariant]], the [[Frobenius norm]] is too. > - The [[Frobenius norm]] is a [[matrix norm]]: it is sub-multiplicative. > - The [[Frobenius norm]] is a special case of the [[Schatten p-norm]] ($p=2$) > [!justification] > $\|\cdot\|_F$ is a valid [[norm]] because the [[Euclidean inner product|Euclidean 2-norm]] is. Also, > ![[CleanShot 2023-09-10 at 21.27.03.jpg]] > [!basicproperties] > - [[spectral bound on Frobenius norm]] > - The Frobenius Norm is **unitarily invariant**, for it depends only on the [[singular values]] of $A$ and [[singular values are 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 > ```