How to solve $$x''+2Ax'+A^2x=0$$ where $A$ is a constant matrix and $x$ is a vector ? I assumed the form was $x=c\cdot \exp(At)$ similar to how it does done for normal second order equations. I got the characteristic equation as $t^2+2At+A^2=0$ with the double root $t=-A$ then I wrote the final solution as:
$$y=c1\cdot\exp(-At)+c2 \cdot t \cdot \exp(-At)$$
Is this correct?