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Math Help - Condition number of simple eigenvalues

  1. #1
    Member Mollier's Avatar
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    Condition number of simple eigenvalues

    Hi, there's a step in the derivation of the condition number that I do not understand.

    Let A\in\mathbb{C}^{n\times n} and let \lambda be a simple eigenvalue of A.
    Let u be a right eigenvector and perturbe such that

    (A+\delta A)(u +\delta u) = (\lambda +\delta\lambda)(u + \delta u).

    I multiply this thing and use the fact that Au=\lambda u to get,

    A(\delta u) +(\delta A)u = \lambda(\delta u) + (\delta\lambda)u.

    Now let v^* be a left eigenvector of A and multiply the above equation from the left. Using the fact that v^*A=v^*\lambda we can write

    v^*(\delta A)u = v^*(\delta\lambda)u,

    hence,

    |\delta\lambda| = \frac{|v^*(\delta A)u|}{|v^*u|}.

    By assuming that ||u||_2=||v||_2=1 we can bound this as

    |\delta\lambda| \leq \frac{||\delta A||_2}{|v^* u|}.

    I hope you are still awake What happens next is confusing me. If \lambda\neq 0 then

    \frac{|\delta\lambda|}{|\lambda|}\leq \frac{||A||_2}{|\lambda||v^*u|}\cdot\frac{||\delta A||_2}{||A||_2},

    and therefore

    cond(\lambda) = \frac{||A||_2}{|\lambda| |v^*u|}.

    I do not understand the (and therefore) part Is this some alternative definition of condition number? Hope someone has the patience to help me out, thanks!
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  2. #2
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    This doesn't look quite right to me either. The condition number of a simple eigenvalue is 1/|v^*u|. What's the reference here?
    Now that I think about it, the quantities given are relative errors. This might be a kind of relative condition number, if such a notion exists.
    Last edited by ojones; May 29th 2011 at 03:57 PM.
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