$ \huge A = U \Sigma V^{\intercal} $ $\large \begin{align} A \in M_{n\times m}\\ U &\in M_{n\times n} \\ \Sigma &\in M_{n\times m}\\ V^{\intercal} &\in M_{m\times m} \end{align} $ [[Singular Value Decomposition]] is that process of decomposing any [[Matrix]] into a [[Orthogonal Matrix|Rotation]] $V^\intercal$, [[Diagonal Matrix|Scale along the cardinal directions]] $\Sigma$, and another [[Rotation Transformation|Rotation]] $U$. Unlike [[Eigendecomposition]], [[Singular Value Decomposition]] can be applied to any $n\times m$ [[Matrix]]. Components: - $U$ is an [[Orthogonal Matrix]] that consists of a [[Orthonormal Basis]] from the [[Eigenvector|Eigenvectors]] $AA^T$ ($AA^T$ is always [[Symmetric Matrix|Symmetric]] therefore its [[Eigenvector|Eigenvectors]] are always [[Orthogonal]]). >[!example]- Example problem: > > >![[../../06 Mind Dump/202504070917|202504070917]]