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Tough Eigenvalue problem

  1. Apr 4, 2010 #1
    1. The problem statement, all variables and given/known data
    [PLAIN]http://img28.imageshack.us/img28/5227/79425145.jpg [Broken]


    3. The attempt at a solution

    I'm not exactly sure how to go about this problem. How do I start?
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Apr 4, 2010 #2

    Dick

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    Re: Eigenvalue

    Start with A^2. A^2(x)=A(A(x)). What's that in terms of lambda?
     
  4. Apr 4, 2010 #3
    Re: Eigenvalue

    Is it lambda^2(x)
     
  5. Apr 4, 2010 #4

    Dick

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    Re: Eigenvalue

    Sure. So that means x is an eigenvector of A^2 with eigenvalue lambda^2, right? The statement for general N>0 follows in the same way.
     
  6. Apr 4, 2010 #5
    Re: Eigenvalue

    So the solution is simply:
    Ax = lambda x
    Therefore A^n x = lambda^n x ?

    Is there something I'm missing?
     
  7. Apr 4, 2010 #6

    Dick

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    Re: Eigenvalue

    That's what the problem is asking you to show, isn't it?
     
  8. Apr 4, 2010 #7
    Re: Eigenvalue

    So what you showed me with A^2 is all I need to answer the question?
     
  9. Apr 4, 2010 #8

    Dick

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    Re: Eigenvalue

    If you understand why it's true, then yes, that's all there is to it. If you want to be formal about proving it you might want to present it as an induction proof.
     
  10. Apr 4, 2010 #9
    Re: Eigenvalue

    Here is my solution:

    A^2(x) = A(A(x))
    A^2(x) = lambda^2(x)

    Therefore x is an eigenvector of A^2 with eigenvalue lambda^2. The general statement for n>0 follows in the same way.

    Is this complete?
     
  11. Apr 4, 2010 #10

    Mark44

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    Re: Eigenvalue

    You should probably use induction rather than say "The general statement for n>0 follows in the same way." That's very vague.
     
  12. Apr 4, 2010 #11
    Re: Eigenvalue

    What do you mean by induction? I've never learned it.
     
  13. Apr 4, 2010 #12

    Mark44

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  14. Apr 4, 2010 #13

    Dick

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    Re: Eigenvalue

    Induction is the formal way to show it. If you've never heard of it and aren't expected to use it the alternative is 'hand waving'. It's easy enough to prove A^3(x)=lambda^3 x using A^2(x)=lambda^2 x. That makes it easy to show A^4(x)=lambda^4 x, etc, etc. So we see it's true for all n. Induction is just the formal way to state 'etc etc'. It's up to you to decide whether the course requires it.
     
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