.

Then to solve $D_t X = AX,$ $\vec{x} (0) = \vec{x_0}$, we have $\vec{x}(t) = \Phi (t) \Phi^{-1}(0) \vec{x_0}$

To solve $D_t X = AX$, we really want $X=e^{At}$. We know $e^x = \sum\limits_{n=0}^{\infty} \frac{x^n}{n!} \Rightarrow e^A = \sum\limits_{... ...rac{A^n}{n!} \Rightarrow e^{At} = \sum\limits_{n=0}^{\infty} \frac{A^n t^n}{n!}$