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Thread: Condition number and the SVD

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    Member Mollier's Avatar
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    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
    Last edited by Mollier; Apr 16th 2011 at 11:09 PM.
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