1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

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


    User Avatar
    Science Advisor

    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>.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook