Complex numbers can be eigenvectors. The matrices in this class are 
carefully chosen so the only eigenvalues are real. For example, symmetric
matrices have only real eigenvalues.

If X1 and X2 are eigenvectors so that A*X1 = L1*X1 and A*X2 = L2*X2 then
for any X = C1*X1 + C2*X2 you can compute A*X by
    A*X = A(C1*X1 + C2*X2) = C1*A*X1 + C2*A*X2 = C1*L1*X1 + C2*L2*X2

Did the parallel lines example; found eigenvectors, eigenvalues, and
characteristic polynomials in scilab.

Worked examples.