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Homework Help: Eigenvectors from complex eigenvalues

  1. Jul 18, 2010 #1
    how does one systematically find the eigenvectors of a 2x2 (or higher) Real matrix given complex eigenvalues?
     
  2. jcsd
  3. Jul 18, 2010 #2

    vela

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    The same way you do it when you have real eigenvalues.
     
  4. Jul 19, 2010 #3

    HallsofIvy

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    For example, the eigenvalues of the matrix
    [tex]\begin{bmatrix}0 & -1 \\ 1 & 0 \end{bmatrix}[/tex]
    are i and - i.

    If < x, y> is an eigenvector corresponding to eigenvalue i then we must have
    [tex]\begin{bmatrix}0 & -1 \\ 1 & 0 \end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix}= \begin{bmatrix}- y \\ x\end{bmatrix}= \begin{bmatrix} ix \\ iy\end{bmatrix}[/tex]

    So we must have -y= ix and x= iy. Since 1/i= -i, those are equivalent. Any such eigenvector is of the form < x, y>= <iy, y>= y<i , 1>.
     
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