# Linear Lyapunov Stability

• Jan 10th 2009, 11:17 AM
ebot
Linear Lyapunov Stability
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.