Positive Definite Matrix Induced by Lorentz Matrix

Question

Assume $G$ is a Lorentzian matrix, which means it has signature $(+,-,\cdots,-)$, and $v$ is a unit timelike vector, i.e. $v^TGv=1$.

So do we have that matrix $2Gvv^TG-G$ is positive definite?

Any advice is helpful. Thank you.