Use the eigenvalue method to find the general solution to the initial value problem:
$x_1' = 3x_1-5x_2$
$x_2' = 5x_1+3x_2$
$x_1(0) = 1$ and $x_2(0) = 4$
I found complex eigenvalues $\lambda=3-5i$ and $\lambda = 3+5i$ which have corresponding eigenvectors $\left[ \begin{array}{cccc} 1\\i \end{array} \right]$ and $\left[ \begin{array}{cccc} 1\\-i \end{array} \right]$. Now I'm not sure how I can write the general solution. Does it involve both eigenvectors?