Suppose we have a linear ODE of the form $\dfrac{dy}{dt}= Ay$. We know that $y= e^{tA} y(0)$ is a solution to the ODE, where $e^{tA}$ represents the matrix exponential. Suppose we also knew that this is stable in the sense of Lyapunov (i.e for any $\varepsilon >0$, there exists a $\delta > 0$ such that if $\vert y(0) \vert < \delta$, then $\vert y(t) \vert < \varepsilon$ for all time $t > 0$). How does that affect the norm of the matrix $e^{tA}$.
OK, so the actual problem is as follows. If a linear ODE $y'= Ay$ is Lyapunov stable and if $B= G^{-1}AG$ where $G$ is an invertible matrix, is $B$ also Lyapunov stable?