Let P and Q be Hermitian positive definite matrices. We prove that for all $\displaystyle x \in \textbf{C}^{n}$ ,

$\displaystyle x^{*}Px \leq x^{*}Qx $ if and only if $\displaystyle x^{*}Q^{-1}x \leq x^{*}P^{-1}x $.

I suppose we need to show that $\displaystyle (Q-P)$ is positive semidefinite iff $\displaystyle (P^{-1} - Q^{-1})$ is also positive semidefinite.

But I am not sure how to proceed.