I need to prove that \(\displaystyle <F''(x)h,h>=0 \Longleftrightarrow F''(x)h=0\) where \(\displaystyle F''(x)\) is the hessian matrix and therefore is PSD \(\displaystyle \Longrightarrow F''(x)=R^TR\) (Cholesky).

Trying to prove the first direction I basically end with \(\displaystyle h^TR^TRh=0\) but can not see why it implies that \(\displaystyle R^TRh=0\)

Can anyone please advice?

TIA,

Best regards,

Giovanni