[[Noteworthy Uses]]:: *[[Noteworthy Uses]]* [[Proved By]]:: *[[Proved By|Crucial Dependencies]]* Intuition:: *[[Intuition]]* Specializations:: [[Singular Value Decomposition of an Operator]], [[Singular Value Decomposition of a Matrix]] Generalizations:: *[[Generalizations]]* ---- - Let [[inner product space]], $\big(W, \langle \cdot,\cdot \rangle\big)$ be $\ff-$[[inner product space|inner product spaces]], where $\ff$ denotes $\rr$ or $\cc$; - Let $T \in$ [[vector space of linear maps between two vector spaces]]; - Let $v \in V$ be arbitrary. > [!theorem] Theorem. ([[Existence of the Singular Value Decomposition of a Linear Map]]) > > Suppose $T$ has *nonzero* [[singular values]] $\{ s_{j} \}_{j=1}^{n}$. Then there exist [[orthonormal]] *lists*[^1] $\{ e_{j} \}_{j=1}^{n} \subset V$ and $\{ f_{j} \}_{j=1}^{n} \subset W$ such that $Tv=s_{1}\langle v,e_{1} \rangle f_{1}+ \dots + s_{n}\langle v,e_{n} \rangle f_{n} $for every $v \in V$. [^1]: Notice the addition of 'nonzero' and change from '[[basis]]' to 'list' in this statement compared to that in the [[existence of the Singular Value Decomposition of an Operator|operator case]]. > [!proof]- Proof. ([[Existence of the Singular Value Decomposition of a Linear Map]]) > Since $T^{\dagger}T \in$ [[vector space of operators on a vector space]], this proof extends from the proof of [[existence of the Singular Value Decomposition of an Operator]]. ---- #### ----- #### References > [!backlink] > ```dataview TABLE rows.file.link as "Further Reading" FROM [[]] FLATTEN file.tags GROUP BY file.tags as 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 > ```