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

I know that A is asymptotically stable.

This is another system
\dot{X}=AX+XA^T where X=xx^T (2)
(i.e. the outer product of little 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 x is a nx1 vector, X and A are nxn matricies.