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Eigenspace and basis of eigenvectors

  1. Oct 17, 2011 #1
    1. The problem statement, all variables and given/known data
    Given the matrix
    0 1 0
    0 0 1
    -3 -7 -5

    Find the eigenspaces for the various eigenvalues
    Prove that there cannot be a basis of R3 consisting entirely of eigenvectors of A


    2. Relevant equations



    3. The attempt at a solution
    The eigenvectors are not a problem, I end up with (λ+3)(λ+1)2 so my eigenvalues are -3 and -1. Substituting in I get [1, -3, 9] and [1, -1, 1]. Now would the eigenspace simply be {[1, -3, 9], [1, -1, 1], [0, 0, 0]} or am I missing some other step?

    Also, how can I prove there cannot be a basis of R3 consisting of eigenvectors of A? Could it just be because there must be n vectors in a basis of Rn?
     
    Last edited: Oct 17, 2011
  2. jcsd
  3. Oct 17, 2011 #2

    Mark44

    Staff: Mentor

    Don't include the zero vector. It can't be a vector in a basis.
    For R3, any basis must contain 3 linearly independent vectors. Since your basis contains only two vectors, these two vectors could not possibly span R3, and so aren't a basis for R3.
     
  4. Oct 17, 2011 #3

    Dick

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    Your characteristic polynomial is wrong. So the eigenvalues and eigenvectors are too. Did you put the correct matrix in the problem statement?
     
  5. Oct 17, 2011 #4

    Mark44

    Staff: Mentor

    Dick makes a good point. After putting in all that effort to find eigenvalues and eigenvectors, you should at least check your work. If x is an eigenvector with eigenvalue [itex]\lambda[/itex], then it should be true that Ax = [itex]\lambda[/itex]x.
     
  6. Oct 17, 2011 #5
    Aaahhh sorry, I missed a 1, it's corrected now
     
  7. Oct 17, 2011 #6

    Dick

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    That's better. Now listen to Mark44 on the basis part of the question.
     
  8. Oct 17, 2011 #7
    Thank you. So I know I can't have the zero vector as part of a basis, but should it be included in the eigenspace? or is the eigenspace juste a basis of the eigenvectors? The terminology confuses me a little.
     
  9. Oct 17, 2011 #8

    Mark44

    Staff: Mentor

    The eigenspace is the set of all linear combinations of the basis vectors. The eigenspace is a vector space, which like all vector spaces, includes a zero vector.

    No one is asking you to list the eigenspace (an impossible task) - just a basis for it.
     
  10. Oct 17, 2011 #9

    Dick

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    Be careful. You have two different eigenspaces here. One corresponding to the eigenvalue -3 and another to the eigenvalue -1. What's a basis for each?
     
  11. Oct 17, 2011 #10
    Oh ok so basis for lambda=-3 is span(1, -3, 9) and for -1 it is span(1, -1, 1).
    Why is it that there can't be a basis for R3 of only eigenvectors?
     
  12. Oct 17, 2011 #11

    Dick

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    It's basically what you said. To span R3 you need three linearly independent eigenvectors. The eigenspaces only give you two.
     
  13. Oct 17, 2011 #12
    Alright thank you so much guys, this was really helpful. Is there some way to +rep or something?
     
  14. Oct 17, 2011 #13

    Dick

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    If you mean a ratings boost, no, we don't have that. Thanks is enough. Very welcome.
     
  15. Oct 17, 2011 #14

    Mark44

    Staff: Mentor

    Same here. Posters don't always say "thank you," but it's appreciated when they do.
     
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