How do you compute the reduced SVD?

Question

I know how to compute the full SVD by hand, but surprisingly, I couldn't find much information on how to compute the reduced SVD by hand. What is the easiest way to do this?

Answer 1

Not necessarily the most efficient way. But this method will work by hand. [From this and the theorems you learn in class, you can easily improve this method, especially in using $A^*A$ versus $AA^*$ and finding $U$).

1. Compute $A^*A$.
2. Find eigenvalues of $A^*A$, and construct $\Sigma$.
3. Find $\Sigma^{-1}$.
4. Compute the eigenvectors of $A^*A$, and construct $V$ and $V^*$ by normalizing these vectors.
5. Find $U$ via $U= AV\Sigma^{-1}$.
6. Write $A= U\Sigma V^*$.

As an example taken from my old lecture notes.