I am confused on how to do this problem, it states:
Find a fundamental set of solutions and put it in general form for the given system.
$x' = \begin{pmatrix} -1/2 & 1\\ -1 & -1/2 \end{pmatrix}x$
I got the eigen values to be $\lambda_1$ = [ (-1/2) + i ] , $\lambda_2$ = [ (-1/2) - i ] so the corresponding eigen vectors are
For $\lambda_1 = v_1 = \begin{pmatrix} 1 \\ i \end{pmatrix}$
For $\lambda_2 = v_2 = \begin{pmatrix} 1\\ -i \end{pmatrix}$
But here is where I get confused on how to write the general solution using Euler's formula for $e^{it} = cost + isint$, thus we have that
$x_1(t) = e^{-t/2}(cost + isint)\begin{pmatrix} 1\\ i \end{pmatrix}$
The answer is below, but how did they get that?
$x_1(t) = \begin{pmatrix} e^{-t/2} cost\\ -e^{-t/2}sint \end{pmatrix} + i\begin{pmatrix} e^{-t/2}sint\\ e^{-t/2}cost \end{pmatrix} = u(t) + iw(t)$
$u(t) = \begin{pmatrix} e^{-t/2} cost\\ -e^{-t/2}sint \end{pmatrix} , w(t) = \begin{pmatrix} e^{-t/2}sint\\ e^{-t/2}cost \end{pmatrix}$