Linear Lyapunov Stability

I have this system:

(1)

I know that is asymptotically stable.

This is another system

where (2)

(i.e. the outer product of little

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 is a nx1 vector, and are nxn matricies.