Examples:: *[[Examples]]* Nonexamples:: *[[Nonexamples]] *Sufficiencies:: *[[Sufficiencies]]* Equivalences:: *[[Equivalences]]* Justifications and Intuition:: *[[Justifications and Intuition]]* Properties:: *[[Properties]]* Constructions:: *[[Constructions|Used in the construction of...]]* Generalizations:: [[Singular Value Decomposition of a Linear Map]] ---- - Let $\ff$ denote $\rr$ or $\cc$; - Let $\mathscr{X} \in$ [[vector space of m-by-n matrices]]. > [!definition] Definition. ([[Singular Value Decomposition of a Matrix]]) > We call the decomposition $\mathscr{X}=U\Sigma V^{\dagger},$ > where $U \in \ff^{m \times m}$ and $V^{\dagger} \in \ff^{n \times n}$ are [[isometric matrix|isometric]] and $\Sigma \in$ [[vector space of m-by-n matrices]] is [[diagonal]], a **singular value decomposition** of $\mathscr{X}$. > \ > In particular, the first $\min \{ m,n \}$ columns of $V$ are the [[right singular vectors]] of $\mathscr{X}$, while the first $\min \{ m,n \}$ of $U$ are the [[left singular vectors]] of $\mathscr{X}$. > ![[Pasted image 20230114112606.png]] > [!warning] > Although we often say 'the' SVD, in general there are *[[uncountably infinite]]* possible SVDs of a given [[matrix]]. For example, if we write $A$ in the [[outer product formulation of the SVD|outer product form]] $A=\sum_{i}^{}\sigma_{i}u_{i}v_{i}',$ > and let $s_{i}=e^{i \phi}$ (e.g., $s_{i}:=-1)$, then another SVD is $A=\sum_{i}^{}\sigma_{i}(s_{i}u_{i})(s_{i}^{*}v_{i}')$ > since $s_{i}s_{i}^{*}=1$. > [!justification] > - [[existence of the singular value decomposition of matrices]] > - [[The SVD.canvas]] ---- #### ---- #### 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 > ```