# Homework Help: Linear system of differential equations with repeated eigenvalues

1. Mar 18, 2013

### richyw

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

2. Relevant equations

n/a

3. The attempt at a solution

The eigenvalues are -1, and $\pm i$. I also can see that the matrix A is already in the form
$$A=\left[\begin{matrix} \alpha & \beta & 0 \\ -\beta & \alpha &0 \\0 & 0 & \lambda\end{matrix}\right]$$ where $\lambda_1=\lambda,\:\lambda_{2,3}=\alpha\pm i\beta$ 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.