Question

refresher on finding eigenvectors

Answer 1

I have the matrix\[\left[\begin{matrix}\alpha & \beta \\ -\beta & \alpha\end{matrix}\right]\]where \(\alpha,\beta \in \mathbb{R}\)

Answer 2

so I get eigenvalues \(\lambda=\alpha\pm i\beta\)

Answer 3

I forget how to get an eigenvector here. For \(\lambda_1=\alpha+i\beta\) an eigenvector is \[v_1=\left[\begin{matrix}1 \\ i\end{matrix}\right]\]

Answer 4

If I remember this correctly, the remaining eigenvector is just the complex conjugate, however I obtain the same result as you do.

Answer 5

I figured it out. I just plugged it into \(AX=\lambda X\) with no problems. don't know why I messed up the first time.