I have the system
$$Y' = \begin{pmatrix} -4 & 4 \\ -8 & 4 \end{pmatrix}Y$$ Using DSolve
DSolve[{x'[t] == -4x[t]+4 y[t], y'[t] == -8x[t] + 4y[t]},{x,y},t] Produces
$$x = c_2 \sin (4 t)+c_1 (\cos (4 t)-\sin (4 t)) \\y = c_2 (\sin (4 t)+\cos (4 t))-2 c_1 \sin (4 t)$$
Finding the Fundamental Matrix, $e^{A t}$, using MatrixExp as
MatrixExp[{{-4t,4t},{-8t,4t}}] Produces
$$\left( \begin{array}{cc} \cos (4 t)-\sin (4 t) & \sin (4 t) \\ -2 \sin (4 t) & \sin (4 t)+\cos (4 t) \\ \end{array} \right)$$
Notice how that perfectly aligns with the DSolve result.
If I use Eigensystem
Eigensystem[{{-4, 4}, {-8, 4}}] This produces
$$ \lambda_i = \pm 4i, v_1 = (1 \pm i, 2) $$
That is a correct result, but I am wondering if we can coax the eigenvectors to be a in a form that can produce the previous two results from the imaginary to real conversion using Euler's Formula.
$$e^{4 i t} = (\cos 4 t + i \sin 4t)\begin{pmatrix} 1 - i \\ 2 \end{pmatrix}$$
Is there a way to peer in to how Mathematica calculated the previous results and make the eigenvectors (since they are not unique) be in a form that I can use this expansion to generate either of the previous two forms?
