Thread: Condition number and the SVD

1. Condition number and the SVD

Hi,

problem;
Let $\displaystyle A$ be an m-by-n matrix of rank $\displaystyle n$.
Evaluate the condition number of,

$\displaystyle M = \begin{pmatrix} I & A\\ A^* & 0 \end{pmatrix}$

I know that the condition number of $\displaystyle M$ is,

$\displaystyle k(A) = \frac{\sigma_{max}(M)}{\sigma_{min}(M)}$ where the $\displaystyle \sigma$ 's are singular values.

If I let $\displaystyle M = U\Sigma V^*$ and try to find the singular values of $\displaystyle M$ by somehow using $\displaystyle A$ it leads me to,

$\displaystyle \Sigma = U^*\begin{pmatrix} I & A\\ A^* & 0 \end{pmatrix}V$.

I've tried some wierd stuff from here, but it does not lead anywhere.

Any hints?
Thanks