this is a question that arose in my mind as I studied Perko's Differential Equations and Dynamical Systems. My intuition says that given an $n\times n$ matrix $A$ with exactly one zero eigenvalue and such that the real parts of every other eigenvalue of $A$ are negative that the dynamical system $x'=Ax$ should converge to some multiple of the eigenvector associated to 0 as the null space of A is spanned by this vector and is one dimensional. I am having trouble proving this fact could anyone point me in the right direction? I have leveraged the Fundamental Theorem to represent the solution to the above system as:
$$ x = Pdiag[B_{j}]P^{-1}[I+N+...+\frac{N^{k-1}t^{k-1}}{(k-1)!}]x_{0} $$
For some change of base matrix P where N is a nilpotent matrix of order $k$ and if $a_{j} = Re(\lambda_{j})$ then
$$ B_{j} = \begin{cases}e^{a_{j}t}\begin{bmatrix} cos(b_{j}t)& -sin(b_{j}t) \\ sin(b_{j}t)& cos(b_{j}t)\\ \end{bmatrix} &\text{if $\lambda_{j}$ is complex}\\ e^{a_{j}t} &\text{otherwise}\\ \end{cases} $$
Not sure where to go next. Any hints?