---- > [!theorem] Theorem. ([[outer product formulation of the SVD]]) > Let $A \in \mathbb{F}^{M \times N}$ and consider the (an) [[Singular Value Decomposition of a Matrix|SVD]] $U \Sigma V'$ of $A$. We have that $A=U\Sigma V'=\sum_{k=1}^{\min (M,N)}\sigma_{i}u_{i}v_{i}',$ > where $u_{i}$ and $v_{i}'$ denote the $i^{th}$ columns of $U$ and $V'$ respectively. > [!proof]- Proof. ([[outer product formulation of the SVD]]) > Really just [[matrix product|matrix multiplication]]. ![[CleanShot 2023-09-17 at [email protected]]] > ![[CleanShot 2023-09-17 at [email protected]]] ---- #### ----- #### 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 > ```