Why is positive semi definiteness a sufficient condition for Lyapunov stability of the Hopfield networks? For a given Hopfeild network : 
$\tau\frac{dx}{dt} = -x + f(Wx + b) $

a general Lyapunov stability function for Hopfeild networks is :

$L(x) = -\frac{1}{2}x^TWx + b^Tx + F(x)$  

where $F(x) = \int^{x} f^{-1}(y)dy$

In the case of linear networks, $f(x) = x$, the Lyapunov function becomes: 

$L(x) = \frac{1}{2}x^T(I - W)x + b^Tx$ 

It is asserted that if $(I-W)$ is positive semi definite, then the $L(x)$ is the Lyapunov function and the system is stable. 
Why is that a sufficient condition?
The fact that it's a Lyapunov function means that
$$
\frac{dL}{dt} = \frac{\partial L}{\partial x} \cdot \frac{dx}{dt} \leq 0
$$
If $I-W$ is positive semidefinite, $L$ is bounded from below by $-\frac{1}{2} b^T (I-W)^{-1} b $.
This means that the network will end up in a minimum of $L$ and cannot keep moving for ever.