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Too many Eigenvectors?

  1. Apr 11, 2012 #1
    Too many Eigenvectors!?

    1. The problem statement, all variables and given/known data
    I have to find the eigenvalues and eigenvectors of:

    -1 2 -2
    1 2 1
    -1 -1 0

    and I can find four eigenvectors I'm not sure how to tell which of my eigenvectors is
    wrong as they all seem to satisfy Av=λv
    (i also checked that they arent simply multiples of each other)

    3. The attempt at a solution
    i used
    det(A-λI)=0
    to get λ=-1,1,1

    then i used the definition Av=λv

    to get the eigenvectors
    [0,1,-1], [1,-1,0], [1,0,-1], [1,-2,1]

    im not sure which of these is wrong and why
     
  2. jcsd
  3. Apr 11, 2012 #2

    Dick

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    Re: Too many Eigenvectors!?

    The eigenvalues of the matrix you showed are not -1,1,1. How did you conclude that? Is there a typo?
     
  4. Apr 11, 2012 #3
    Re: Too many Eigenvectors!?

    yes it was a typo sorry
    the matrix was meant to be

    -1 -2 -2
    1 2 1
    -1 -1 0
     
  5. Apr 12, 2012 #4

    Dick

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    Re: Too many Eigenvectors!?

    Writing v1=[1,-1,0] and v2=[0,1,-1], v1 and v2 are both independent eigenvectors corresponding to the eigenvalue 1. And you are allowed to have two of those since the eigenvalue 1 has multiplicity 2. The other two vectors you wrote are v1+v2 and v1-v2. There are lots more eigenvectors corresponding the eigenvalue 1 as well, any linear combination of v1 and v2 will do. What you are missing is the eigenvector corresponding to the eigenvalue -1.
     
    Last edited: Apr 12, 2012
  6. Apr 19, 2012 #5
    Re: Too many Eigenvectors!?

    thanks! I hadn't realised that linear combinations would be solutions. I now have a correct ( I think) set of vectors

    v1=[-1 1 0]
    v2=[-1 0 1]
    v3=[2 -1 1]

    :)
     
  7. Apr 19, 2012 #6

    HallsofIvy

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    Re: Too many Eigenvectors!?

    One of the very first things you should have learned about eigenvectors is there is NOT a single unique eigenvector corresponding to a given eigenvalue. In fact, the set of all eigenvectors corresponding to a given eigenvalue form a subspace which necessarily contains linear combinations.
     
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