Question

Diagonalization of a matrix - finding the eigenvectors

Answer 1

I have a matrix to diagonalize but I seem to get a different result for two of the eigenvectors.

This is the matrix:
\[\left[\begin{matrix}1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0\end{matrix}\right]\]
So, the eigenvalues are 1, -i and i.

I got the correct answer for the eigenvector for a = 1 but there is something wrong about the other two. What I have is this:
\[\left[\begin{matrix}1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0\end{matrix}\right] \left(\begin{matrix}x1 \\ x2 \\ x3\end{matrix}\right) = i\left(\begin{matrix}x1 \\ x2 \\ x3\end{matrix}\right) \] so \[ \begin{matrix} x_1 = ix_1 \\ -x_3 = ix_2 \\ x_2 = ix_3 \end{matrix}\]
The answer that I am supposed to get after normalization is \[\frac{ 1 }{ \sqrt{2} }\left(\begin{matrix}0 \\ i \\ 1\end{matrix}\right)\]

Answer 2

and \[ \frac{ 1 }{ \sqrt{2} }\left(\begin{matrix}0 \\ -i \\ 1\end{matrix}\right) \]for a = -i