The matrix is
alpha 0 1
0 Beta 0
-1 0 alpha
The eigenvalues of this is Beta, alpha±i which is fine.
I don't however understand how the eigenvectors end up being
For lambda = beta, (0,1,0)
For lambda = alpha ± i, (1,0,0),(0,0,1).
Any help would be appreciated as I have an exam tomorrow!