A diagonalizable system $A$ is internally stable, or state space stable, if and only if for every state $\mathbf{x}$, the system’s natural response (i.e. with no input) results in a bounded state trajectory.
$$ \forall \mathbf{x}(t), \space \mathbf{\dot{x}}=A\mathbf{x}, \space \exists L \in \R\space \vert \space \forall t \in \R ^{\geq 0}, ||\mathbf{x}(t)|| \leq L \\
\hspace{21pt} \forall \mathbf{x}[i], \space \mathbf{x}[i+1]=A\mathbf{x}, \space \exists L \in \R\space \vert \space \forall i \in \Z ^{\geq 0}, ||\mathbf{x}[i]|| \leq L $$
In linear, time-invariant systems, an internally stable system will have all state trajectories converge at $\mathbf{0}$. There are 3 major classifications of internal stability,
The internal stability of a system $A$ is entirely characterized by its eigenvalues.
$$ \mathbf{x}(t) = e^{At} \mathbf{x}(0) = \sum _{k=1} ^{\ell} \mathbf{c}k e^{\lambda_k t} \hspace{15pt} \text{ and } \hspace{15pt}\mathbf{x}[i] = A^{i} \mathbf{x}[0] = \sum{k=1} ^{\ell} \mathbf{c}_k \lambda ^{i}_k $$
Since the eigenvalues are exponents, the end behavior of $||\mathbf{x}(t)||$ depends on the sign of $\text{Re}\space \{\lambda _i\}$.
$$ \lambda_i = \sigma _i \pm j \omega_i\rightarrow e^{\lambda_i t}=e^{\sigma_i t} (\cos (\omega_i t) \pm j\sin(\omega_i t)) \\ $$
The $\pm$ accounts for the fact that complex roots to a real system come in conjugate pairs (i.e. if $z$ is a root, so is $z^*$). Given this formulation, we can describe the stability of the system in terms of $\sigma _i$.
$$ \begin{cases} \text{All } \sigma _i < 0 \hspace{83pt} \rightarrow \text{Stable} \\ \text{Any } \sigma _i > 0 \hspace{78pt}\rightarrow \text{Unstable} \\ \text{All } \sigma _i \leq 0 \text{ and some } \sigma _k = 0 \hspace{10pt} \rightarrow \text{Marginally Stable or Unstable} \end{cases} $$
Since the eigenvalues are the bases, the end behavior of $||\mathbf{x}(t)||$ depends on the size of $\text{Re} \{\lambda _k\}$