Suppose the matrix A with real entries has complex eigenvalues $\lambda = \alpha + i\beta\,$ and $\overline{\lambda} = \alpha - i\beta\,$. Suppose that $Y_0 = (x_1 + iy_1, x_2 + iy_2)$ is an eigenvector for the eigenvalue $\lambda$. Show that $\overline{Y_0} = (x_1 - iy_1, x_2 - iy_2)$ is an eigenvector for the eigenvalue $\overline{\lambda}$. In other words, the complex conjugate of an eigenvector for $\lambda$ is an eigenvector for $\overline{\lambda}$.
This is what I have:
$Av = \lambda v$ but lost with notations. Can somebody explain how to do this?