---- Let $A \in$ [[vector space of m-by-n matrices]], where $\ff=\rr$ or $\ff=\cc$. > [!definition] Definition. ([[spectral matrix norm]]) > The **spectral norm** of $A$ is defined to be $\|A\|_{2}:=\max_{x:\|x\|_{2}=1}\|Ax\|_{2}=\max_{x \neq 0} \frac{\|Ax\|_{2}}{\|x\|_{2}}=\sigma_{1}.$ > Here, $\|\cdot\|_{2}$ when applied to $x \in \mathbb{F}^{n}$ denotes the [[Euclidean inner product|Euclidean 2-norm]]. $\sigma_{1}$ denotes the largest [[singular values|singular value]] of $A$. > \ > **Remark.** The smallest [[singular values|singular value]] also corresponds to an optmiazation (slide 3.4 4 eecs 551 chapter 3 [[TODO]]) > [!note] > The [[spectral matrix norm]] is the [[matrix norm]] [[induced matrix norm|induced]] by the [[Euclidean inner product|Euclidean vector norm]] ([[Lp-norm|L2 norm]]) $\|\cdot\|_{2}$. > [!intuition] > The spectral norm is thought of as finding the direction in which the maximum amount of 'stretching' is done by a [[linear operator]]. This is intuitive because $\sigma_{1}$ is associated to the largest [[eigenvalue]] of the 'positivization of $A , $A'A$, and [[positive semidefinite operator]]s are often though of as 'stretching' space. > [!justification] > The equalities above are all true by [[largest singular value tightly maximizes power]]. > [!basicproperties] > The spectral norm is **unitarily invariant**, for it depends only on [[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 > ```