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Reminder on how to find eigenvectors

  1. May 3, 2012 #1
    Ok everybody, it's been awhile since I've taken linear algebra. I need some help dusting off the cobwebs. (I'm trying to follow this in a paper; this isn't a homework question, but I'll be glad to move it...)

    Let's say I have a matrix M = \begin{bmatrix}
    -σ & σ & 0 \\[0.3em]
    ρ & -1 & 0 \\[0.3em]
    0 & 0 & -1
    \end{bmatrix} \

    Ok, now I want to find the eigenvectors (I've already found the eigenvalues.

    These are λ_1=-1 and λ_2,3 = -(σ+1)/2 +/- ( (σ+1)^2 -4σ(1-ρ))^.5

    Now, to find the eigenvectors, I solve these equations:

    (-σ-λ)e_1 +σ e_2 =0
    ρ e_1 +(-1-λ) e_2 =0
    (-1-λ) e_3 =0

    Now, I see that the eigenvector for λ_1 is ( 0 0 1).

    Now I'm stuck. How do I find the eigenvectors for the other two?

    The result in the paper I'm reading says

    λ_2,3 = (σ σ+λ_2 0 ) , (σ σ+λ_3 0 )

    I forget how to do this....
  2. jcsd
  3. May 3, 2012 #2


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    Science Advisor

    Essentially, just the way you did the first part, just much messier algebra!
  4. May 3, 2012 #3

    So, (-σ-λ)e_1 +σ e_2 =0

    Looking at this and getting (σ+λ)e_1 =σ e_2, I would think the eigenvector is (σ+λ σ) not (σ+λ σ).
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