Complex eigenvalues

I extended a model, calibrated it and solved it with Dynare. The Blanchard Kahn conditions are satisfied, but I have 2 conjugated complex eigenvalues.

I have a problem understanding how this is possible. If X(t) = lambda * X(t-1) where lambda is the matrix where the eigenvalues are on the diagonal, how can I still have the variable in X(t) which is related to the complex eigenvalue since I am multiplying its lag by a complex eigenvalue?

Please have a look at Villemot (2011): Solving rational expectations models at first order: what Dynare does. The eigenvalues reported are generalized eigenvalues resulting from the generalized Schur/QZ decomposition. They do not need to be real.

Indeed, I was confusing the matrix that included the eigenvalues with the policy function.

Thank you for your answer.