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Proving a property of eigenvalues and their eigenvectors.

  1. May 24, 2014 #1
    1. The problem statement, all variables and given/known data

    I am asked to prove that if λ is an eigenvalue of A then λ + k is an eigenvalue of
    A + kI.

    3. The attempt at a solution

    ## A\vec{v}=\lambda\vec{v} ##

    ## (A+kI)\vec{v}=\lambda\vec{v} ##
    ## A\vec{v}+k\vec{v} = \lambda\vec{v} ## → ## A\vec{v} = \lambda\vec{v} - k\vec{v} ##
    ## A\vec{v} = (\lambda-k)\vec{v} ## however, this is not λ+k
    I must be missing something pretty obvious, but i can't see what it is...
     
  2. jcsd
  3. May 24, 2014 #2

    Matterwave

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    Science Advisor
    Gold Member

    Your second step is the error. You have assumed in that step that lambda is an eigenvalue of A+kI in that step.
     
  4. May 24, 2014 #3
    Ahh thank you! I didn't even realise that, im so use to denoting the eigenvalues lambda..
     
  5. May 24, 2014 #4

    Zondrina

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    Homework Helper

    If ##\lambda + k## is an eigenvalue of ##A + kI##, then:

    ##Av = \lambda v##
    ##Av + (kI)v = \lambda v + (kI)v##
    ##(A + kI)v = (\lambda + kI)v##
     
    Last edited: May 24, 2014
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