I have this system:
$\displaystyle \dot{x}=Ax$ (1)

I know that $\displaystyle A$ is asymptotically stable.

This is another system
$\displaystyle \dot{X}=AX+XA^T$ where $\displaystyle X=xx^T$ (2)
(i.e. the outer product of little $\displaystyle x$

I know that system (1) is asymptotically stable iff system (2) is asymptotically stable.

How do I show that system (2) is stable (not necessarily asymptotically stable) iff there is a linear Lyapunov function proving it's stability. Note I said "linear Lyapunov function", not "quadratic Lyapunov function".
Also note $\displaystyle x$ is a nx1 vector, $\displaystyle X$ and $\displaystyle A$ are nxn matricies.