Originally Posted by

**AlexP** I would appreciate it if someone could just verify whether this works or not.

Prove that if T is positive definite, then T^{-1} is positive definite.

Since $\displaystyle \langle T(x),x \rangle > 0$ and $\displaystyle T(T^{-1}(x))=x$, we have $\displaystyle 0 < \langle T(x),x \rangle = \langle T(x),T(T^{-1}(x)) \rangle = \langle T^{-1}(T(x)),T^{-1}(T(T^{-1}(x))) \rangle$

$\displaystyle = \langle x,T^{-1}(x) \rangle = \langle T^{-1}(x),x \rangle$. The last equality is true since $\displaystyle \langle T(x),x \rangle$ is real, and also proves that $\displaystyle T^{-1}$ is self-adjoint, and thus is positive definite.

Thanks.