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Differentiation Map of a Complex Transformation

  1. Nov 11, 2012 #1
    1. The problem statement, all variables and given/known data

    Find the eigenvectors and eigenvalues of the differentiation
    map C1(R) -> C1(R) from the vector space of differentiable functions
    to itself.

    2. Relevant equations



    3. The attempt at a solution

    Hi, I'm not entirely sure how to go about this, because would the differentiation map of this be [(1,0),(0,1)] since its from it to itself? Thanks for the help in advance.
     
  2. jcsd
  3. Nov 11, 2012 #2

    tiny-tim

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    hi nautolian! :smile:
    an eigenvector is an element f of C1(R) such that Df is a scalar times f :wink:
     
  4. Nov 12, 2012 #3
    High, sorry I'm still not really sure where to go with this. I mean I understand that Df=(lambda)f, but in the terms of C^1(R) does this mean that the derivative of the complex number a+bi is the same as the eigenvalue times the vector? Sorry, I'm still somewhat lost. Thanks for your help though.
     
  5. Nov 12, 2012 #4

    Dick

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    I don't think this has anything to do with complex numbers. C^1(R) usually just means differentiable real functions. You need to solve the differential equation f'(x)=λf(x).
     
  6. Nov 14, 2012 #5
    okay, would that mean that there are infinite eigenvalues with associated iegenvectors equal to A*exp(lambda*x)? where A is a constant? Sorry I'm still unclear about this. Thanks for the help though!
     
  7. Nov 14, 2012 #6

    Dick

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    Yes, that's it.
     
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