Lyapunov equation and eigenvalues

It is often observed that the linear, constant coefficient, ODE system,

$\displaystyle

\dot{x}=Ax

$,

is asymptotically stable iff every eigenvalue of A has a negative real part. I've also just learned that the above system is asymptotically stable iff there exists a symmetric positive definite matrix P and a symmetric negative definite matrix Q which satisfy the Lyapunov equation

$\displaystyle

A^TP+PA=Q

$.

At least, I think I've got that right.

How do these truths relate? Can the eigenvalues of A be shown to have negative real parts by the properties of P and Q in the Lyapunov equation?