SVD and PCA

SVD is intimately related to eigendecomposition

• $A$ is square and symmetric: $U$ and $V$ have equal columns up to possible sign differences
• if $A$ is positive semi-definit, then the SVD is equal to the eigendecomposition
• Squares of $A$: $A{A}^T\in {\mathbb{R}}^{n\times n}$ as well as $A^{T}A\in {\mathbb{R}}^{m\times m}$:

convention: ${\sigma }_r=0$ for $min\{n,m\}<r\le max\{n,m\}$