## Franklin's Notes

### Reduced SVD

The reduced singular value decomposition is a variation of the singular value decomposition in which "silent rows" of $V^\ast$ are removed. That is, if the matrix $A$ has $r$ singular values equal to zero, then $A$ can be decomposed in the form where $\hat U$ is $m\times r$, $\hat\Sigma$ is $r\times r$, and $\hat V$ is $n\times r$. This decomposition is formed from the SVD by removing the columns of $U$ and $V$ that are "annihilated" by the zero rows of $\Sigma$, and then removing these zero rows from $\Sigma$. The standard signular value decomposition is sometimes called a full SVD.