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Eigenvector with Complex Eigenvalues - What am I doing wrong?

  1. Apr 30, 2010 #1
    1. The problem statement, all variables and given/known data

    2. Relevant equations
    Conjugate of a complex number
    Matrix reduction

    3. The attempt at a solution

    My attempt is bordered. Sorry about the quality.

    So I'm not sure what I'm missing. I use the exact same method that I use for normal eigenvectors, just with complex numbers in the mix.
    Last edited: Apr 30, 2010
  2. jcsd
  3. Apr 30, 2010 #2


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    You messed up in the last step and swapped x1 and x2. If you let x2=1, then your equation gave you x1=-1/2+1/2 i. In the vector, however, you have the two values in the other order.
  4. Apr 30, 2010 #3
    I can't quite read all of your work, but if you subtract [tex]\lambda[/tex] along the diagonal, you get:

    [tex] \bmatrix
    1 + i & 1 \\
    -2 & -1 + i \\
    \endbmatrix x = 0 [/tex]

    The rows of this matrix are multiples (-1 + i) of each other, so you can use either row to find the eigenvector:

    [tex] x_1 (1 + i) + x_2 = 0 [/tex]
    [tex] -x_1 (1 + i) = x_2 [/tex]

    so your vector is (1, -1 - i). This happens to not be one of your choices, but you can multiply it by i to obtain (i, 1 - i).
    Last edited: Apr 30, 2010
  5. Apr 30, 2010 #4
    @vela Thanks. I wasn't really thinking it through at the end.
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