Finding basis of complex 3 - vectors of a matrix and eigenvalues

Question

Find a basis of complex 3-vectors and complex eigenvalues for the matrix $$ \begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix} $$

I found the eigenvalues to be $\lambda = 1, \lambda = -\frac12i\big(\sqrt{3} - i\big), \lambda = \frac12i\big(\sqrt{3}+i\big)$

Not sure how to go about finding the basis vectors though. Perhaps its something simple like the three vectors (columns) $$\begin{bmatrix} 0 & 0 & -i^2 \\ -i^2 & 0 & 0 \\ 0 & -i^2 & 0 \end{bmatrix} $$ But Im really not sure.

Answer 1

For each of your $\lambda$'s, find a non-null vector $v$ such that$$\begin{bmatrix}0&0&1\\1&0&0\\0&1&0\end{bmatrix}.v=\lambda v.$$Those $3$ vectors will form the basis that you're after.