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Linear system of differential equations with repeated eigenvalues

  1. Mar 18, 2013 #1
    1. The problem statement, all variables and given/known data
    [tex]X'=AX[/tex][tex]A=\left[\begin{matrix} 0 & 1 & 0 \\ -1 & 0 &0 \\0 & 0 & -1\end{matrix}\right][/tex]

    2. Relevant equations

    n/a

    3. The attempt at a solution

    The eigenvalues are -1, and [itex]\pm i[/itex]. I also can see that the matrix A is already in the form
    [tex]A=\left[\begin{matrix} \alpha & \beta & 0 \\ -\beta & \alpha &0 \\0 & 0 & \lambda\end{matrix}\right][/tex] where [itex]\lambda_1=\lambda,\:\lambda_{2,3}=\alpha\pm i\beta[/itex] So I don't see the point really in computing the eigenvectors because this is already in canonical form isn't it? so I don't need to find T that would change the original matrix into its canonical form. So I think that the solution to X'=AX would just be Y(t). I have NO IDEA how to find Y(t) though. My book doesn't show the steps.
     
  2. jcsd
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