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Eigenvectors & normalising

  1. Jan 5, 2007 #1

    kel

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    1. The problem statement, all variables and given/known data
    Hi, I'm going over an old exam paper as part of my revision for upcoming exams (joy !! ok, maybe not :yuck: )

    Anyway, I've gotten myself a bit lost here and would appreciate some guidance.
    Here we go:

    Compute the eigenvalues & eigenvectors of the following matrix. Normalise the 2 eigenvectors.

    Matrix = (3 1+i)
    (1-i 2)

    2. Relevant equations



    3. The attempt at a solution

    So far I have the 2 eigenvalues being +1 and +4, but I'm having trouble with the next bit, I think that the eigenvectors will come out as complex numbers,

    so far I have reduced to two equations:

    2x + (1+i)y = 0
    (1-i)x + y = 0

    ie 2x + (y+iy) = 0 (so, 2x = -y-iy)
    and x-ix + y = 0 (and y = -x+ix)

    Is this correct?? if not why? and where do I go from here?

    Also, I don't get the idea of normalisation (next part of question), I understand that it has something to do with setting the vector(i think) to 1, but I'm not sure. My lecturers notes aren't the easiest to follow outside of his lectures, so if you could walk me through it a bit I'd be very grateful.

    Thanks
    Kel
     
  2. jcsd
  3. Jan 5, 2007 #2

    cristo

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    Staff Emeritus
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    Is this a problem? You are working in a complex vector space, and so the eigen-vectors are entitled to have complex entries!
    I've not checked your work, but you look to be doing the correct method. Why not try solving eqns (1) and (2) for x and y?
    Normalised means that the vectors have unit length. Once you have the eigenvector, v, say, then the normalised vector is v/|v|
     
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