Condition number and the SVD

**Mollier** · April 6th 2011, 11:34 PM

Hi,

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

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

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

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

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

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

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

Any hints?
Thanks

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