Examples:: *[[Examples]]* Nonexamples:: *[[Nonexamples]]* Constructions:: [[singular value decomposition of a matrix]] Generalizations:: *[[Generalizations]]* Justifications and Intuition:: *[[Justifications and Intuition]]* ---- - 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. > [!definition] Definition. ([[Singular Value Decomposition of a Linear Map]]) > The **Singular Value Decomposition** of $T$ is the following representation of $Tv$ in terms of two specific [[orthonormal|orthonormal lists]] $\{ e_{j} \}_{j=1}^{n} \subset V$ and $\{ f_{j} \}_{j=1}^{n} \subset W$: $Tv=s_{1}\langle v,e_{1} \rangle f_{1} +\dots + s_{n}\langle v,e_{n} \rangle f_{n} , $where $\{ s_{j} \}_{j=1}^{n}$ denote the *nonzero* [[singular values]] of $T$. **Remark**. The primary difference between this definition and that for [[Singular Value Decomposition of an Operator]] is the addition of the term *nonzero* and change from [[orthonormal basis]] and [[orthonormal|orthonormal list]]. ---- #### ---- #### 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 > ```