Exercise 7

Suppose that $A$ is a square $m\times m$ matrix with singular value decomposition $A=U\Sigma V^T$.

1. If $A$ is invertible, find a singular value decomposition of $A^{-1}$.

2. What condition on the singular values must hold for $A$ to be invertible?

3. How are the singular values of $A$ and the singular values of $A^{-1}$ related to one another?

4. How are the right and left singular vectors of $A$ related to the right and left singular vectors of $A^{-1}$?