I saw an explanation as follows.
Suppose a linear system dx(t)/dt=Ax(t) (1)
1, Re λi 0 for any Q>0.
2,V(x):=x'Px is a Lyapunov function with Lie derivative dotV(x):=xTQx where dotV is negative definite.
3,By the Lyapunov's second method, 1 and 2 lead to the claim that 'Re λi