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!
the original answers came from my lecturer. the question was about plotting phase portraits of a 3D linear vector field and then arguing why they are topologically equivalent, with the results being alpha±i drawn in the X-Z plane, and beta drawn in the Y plane, so i am assuming what you got was what he meant by the X-Z plane. I had another go by hand and I got the same results as you did. I am not sure the of the relevance of the complex number however as it seems to be disregarded in order to get the same results as he did