Is there more than one possibility for eigenvectors of a single eigenvalue

[mkerikss]

I guess this is best explained with an example. The matrix (0 -1) has the eigenvalues
------------------------------------------------------------------ (1 0)
i and -i. For -i we obtain ix₁-x₂=0 and x₁+ix₂=0. I got a corresponding eigen vector (1 i), but when I controlled this result with wolfram alpha it gave me the vector (-i 1). By inserting these values in the equations both of these give 0, as they should, so my guess is they can both be used as the eigenvector for the eigenvalue -i, but I'm not sure though, as I haven't come across this kind of situation before. If somebody could help it would be most appreciated.

 

[vr88]

Note that those two vectors are scalar multiples of each other, and eigenvectors are determined up to scalar multiples.

 

[mkerikss]

Ok, I noticed that now, if i multiply the first one with -i I get the second one and vice versa. Thank you