I have a two planar systems $X'=AX$ and $Y'=BY$ both with eigenvalues $\pm i \beta$. I know I can use the fact that there exists two linear transformations T and S such that
$$T^{-1}AT=S^{-1}BS=\begin{bmatrix} 0 & \beta \\ -\beta & 0 \end{bmatrix}= C$$
where C is in canonical form. Now to show that these two systems are conjugate I wish to use the homeomorphism $h = S \circ T^{-1}$ but I have been stuck on how to show
$$h(\phi_t^A(x_0)) = \phi_t^B(h(x_0))$$
Is my first step the right way to approach this problem? I am not sure why proving the last equality is so hard for me
$$ ST^{-1} e^{At} X_0 = e^{Bt} S T^{-1} X_0 $$
The main thing is for constant matrices, if $P^{-1} U P = W$ then $P^{-1} e^{Ut} P = e^{Wt}$