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Finding the eigenvectors of a 2nd multiplicity engenvalue

  1. Feb 17, 2013 #1
    1. The problem statement, all variables and given/known data
    I'm given this matrice 2 1 0
    1 2 0
    0 0 3
    and I need to find it's eigenvectors



    2. Relevant equations



    3. The attempt at a solution

    So I get the eigenvalues to be 1,3,3 with 3 being the one with multiplicity of 2. For the eigenvector for 1 I get 1,-1,0 and for 3 I get 1,1,1 but here is the problem. For the other eigen vector for 3 the answer can be anything that satisfies x1=x2
    x2=x1
    x3=x3

    so can something like 1,1,2 be the answer?
     
  2. jcsd
  3. Feb 17, 2013 #2

    Dick

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    Sure it could. Try it out if you are unsure. There are a lot of choices for specifying the eigenvectors. Any linear combination of 1,1,1 and 1,1,2 will also be an eigenvector with eigenvalue 3.
     
  4. Feb 17, 2013 #3
    But wouldn't that mean there is no one definitive eigenbasis?
     
  5. Feb 17, 2013 #4

    Dick

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    Sure it would. There never is. You have a two dimensional space of eigenvectors with eigenvalue 3. There are lots of ways to choose a basis. I wouldn't call any of them 'definitive'.
     
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