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!