Eigenvalues of $6 \times 6$ matrix?

Question

Which of {$\pm1,\pm i$} are the eigenvalues of matrix A, $$A=\begin{pmatrix} 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ \end{pmatrix}$$

My working suggested $+1$ and $+i$. Am I correct?

Answer 1

Hint: For this particular matrix, you have eigenvectors of the form $(1,0,0,1,0,0)$ and $(1,0,0,-1,0,0)$ etc.

Answer 2

Your matrix exchanges $x_1$ with $x_4$, $x_2$ with $x_5$, and $x_3$ with $x_6$. So if $x_1$ and $x_4$ are the same, then you have an eigenvalue of $1$ with that eigenvector. The same holds for the other two options.

What happens when $x_i = -x_{i+3}$?. Checking that case will account for six independent eigenvectors, so any other is a combination of those.