An operator bound on some state and the state operator

Question

Thanks in advance.

I'm struggling with some statement in a paper. It claims as follows,

For a p.s.d. operator $H:\mathbb{R}^d\to\mathbb{R}^d$ with an inverse and some $v\in\mathbb{R}^d$, such that

$v^{T}Hv\le\gamma$

for some constant $\gamma > 0$,

then $vv^T\preceq \gamma{H^{-1}}$

I cannot see it is obvious. Please help.

Answer 1

For any $x\in\mathbb{R}^d$, we have the following

$x'vv'x=v'xx'v=v'H^{1/2}H^{-1/2}xx'H^{-1/2}H^{1/2}v$

$\le trace(H^{-1/2}xx'H^{-1/2})v'Hv=x'H^{-1}x*v'Hv\le\gamma x'H^{-1}x$

The conclusion follows by the arbitrariness of $x$.