I was trying to solve the eigenvalues and vectors of the following matrix
$$\begin{bmatrix} 0& 0&1 \\ 1& 0 & 0\\ 0&1 & 0 \end{bmatrix}. $$
I found the eigenvalues to be $\omega_1 = \frac{-1+i\sqrt{3}}{2}$, $\omega_2 = \frac{-1-i\sqrt{3}}{2}$ and $\omega_3 = 1.$
I easily found the eigenvector for the $\omega_3$. but I got stuck on the complex eigenvector, specifically solving the linear system resulting from the eigenvector-eigenvalue equation. For example, I found the following for $\omega_1$:
$$\begin{bmatrix} -\omega_1& 0&1 \\ 1& -\omega_1 & 0\\ 0&1 & -\omega_1 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2\\ x_3 \end{bmatrix}$$
the resulting system is then: \begin{align*} \left(\frac{1-i\sqrt{3}}{2}\right)x_1+x_3&=0 \\ x_1+\left(\frac{1-i\sqrt{3}}{2}\right)x_2&= 0\\ x_2+\left(\frac{1-i\sqrt{3}}{2}\right)x_3&= 0. \end{align*} I substituted $1$ for $x_3$ and obtained the following normalized vector
$$\frac{1}{\sqrt{3}}\begin{bmatrix} \frac{-1-i\sqrt{3}}{2}\\ \frac{-1+i\sqrt{3}}{2}\\ 1\end{bmatrix}$$
however in the solution manual of the book the answer is :
$$\frac{1}{\sqrt{3}}\begin{bmatrix} 1\\ \frac{-1-i\sqrt{3}}{2}\\ \frac{-1+i\sqrt{3}}{2}\end{bmatrix}. $$
I am not sure what the mistake I did. I tried dotting this with what I found to see if mine is in the same span, but the dot product resulted in a zero value. Could someone show what I am doing wrong here. I suspect it has to do with the way I solved the system of equation since I am not familiar with complex systems.
What makes you think that there is an error? The second vector is the first one multiplied by $\frac{-1+\sqrt3\,i}2$. There is no contradiction here.