Consider the linear ODE system:
$\dot x = \begin{pmatrix} 1 & -1 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{pmatrix}x, x(t_0)=x_0 $
(i) Calculate the general solution
(ii) Calculate the particular solution when $x(0)=(0,0,1)^T$
(iii) Calculate the restriction of the dynamics to the unstable subspace $E^u$.
I believe I have the general solution for (i) with
$e^{tA}=e^t\begin{pmatrix} 1 & -t & t-t^2 \\ 0 & 1 & 2t \\ 0 & 0 & 1 \end{pmatrix}$
(my working is quite long but I could add it if necessary)
so the general solution would be
$x(t)= e^t\begin{pmatrix} 1 & -t & t-t^2 \\ 0 & 1 & 2t \\ 0 & 0 & 1 \end{pmatrix}x(0)$?
for (ii) I replaced x(0) with the conditions to get:
$x(t)= e^t\begin{pmatrix} 1 & -t & t-t^2 \\ 0 & 1 & 2t \\ 0 & 0 & 1 \end{pmatrix}\begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}=e^t\begin{pmatrix} t-t^2 \\ 2t \\ 1 \end{pmatrix}$?
but I'm really not sure where to go from here, any help?