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Eigenvectors and Eigenspace

  1. Jul 2, 2011 #1
    1. The problem statement, all variables and given/known data
    Im looking at finding the eigenvectors of a matrix but also a basis for the eigenspace

    A = [ 6 16 ]
    [ -1 -4 ]

    lambda = 4
    lambda = -2


    2. Relevant equations
    (A - lambda I ) v = 0


    3. The attempt at a solution

    So with the above equation I get:

    for lambda = 4

    [ 6 - 4 16 ] [ v1 ] = [ 0 ]
    [ -1 -4 - 4 ] [ v2 ] [ 0 ]

    so

    2 v1 + 16 v2 = 0
    -v1 - 8v2 = 0

    so v1 = 8v2

    and the basis for the eigenspace is span [ 8 ]
    [ 1 ]

    First is that right? because when I put it into an eigenvector calculator on the web it gives me
    -8 instead of 8 but I cant see how I could get to that.

    Second if this is the basis for the eigenspace then how can I find the eigenvectors for the eigenvalue?

    thanks,
     
  2. jcsd
  3. Jul 3, 2011 #2

    ehild

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    You made a sign error. v1=-8v2

    ehild
     
  4. Jul 3, 2011 #3
    Oh yeah thats right thanks.

    Is it as simple as the vector is:

    [-8v2]
    [v2]

    and the eigenspace is: span

    [-8]
    [1]
     
  5. Jul 4, 2011 #4
    Sorry I realised this should have been posted in the calculus section would it be possible to have it moved?

    I think what I have put above for the eigenspace is correct? But what about the eigenvector I cant seem to understand what this is.
     
  6. Jul 4, 2011 #5

    ehild

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    (-8,1) multiplied by any number is an eigenvector. You need to find the other one, which belongs to the other eigenvalue lambda=2.
    The two eigenvectors are the basis of the "eigenspace". You can choose the normalised vectors as basis.

    ehild
     
  7. Jul 5, 2011 #6
    Oh thanks. Do I need to normalise the vectors or is it fine to just find the two vectors and give that?
     
  8. Jul 5, 2011 #7

    ehild

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    You do not need to normalize in principle.

    ehild
     
  9. Jul 5, 2011 #8
    ok thanks.
     
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