# Positive definiteness of an inverse

• December 27th 2011, 08:24 PM
AlexP
Positive definiteness of an inverse
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 $\langle T(x),x \rangle > 0$ and $T(T^{-1}(x))=x$, we have $0 < \langle T(x),x \rangle = \langle T(x),T(T^{-1}(x)) \rangle = \langle T^{-1}(T(x)),T^{-1}(T(T^{-1}(x))) \rangle$
$= \langle x,T^{-1}(x) \rangle = \langle T^{-1}(x),x \rangle$. The last equality is true since $\langle T(x),x \rangle$ is real, and also proves that $T^{-1}$ is self-adjoint, and thus is positive definite.

Thanks.
• December 28th 2011, 12:33 AM
Drexel28
Re: Positive definiteness of an inverse
Quote:

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 $\langle T(x),x \rangle > 0$ and $T(T^{-1}(x))=x$, we have $0 < \langle T(x),x \rangle = \langle T(x),T(T^{-1}(x)) \rangle = \langle T^{-1}(T(x)),T^{-1}(T(T^{-1}(x))) \rangle$
$= \langle x,T^{-1}(x) \rangle = \langle T^{-1}(x),x \rangle$. The last equality is true since $\langle T(x),x \rangle$ is real, and also proves that $T^{-1}$ is self-adjoint, and thus is positive definite.

Thanks.

This looks fine, but it may have been easier to just note that being positive definite is equivalent to having strictly positive eigenvalues, and since the eigenvalues of $T^{-1}$ are the reciprocals of the eigenvalues of $T$ everything quickly follows.
• December 28th 2011, 12:56 AM
ILikeSerena
Re: Positive definiteness of an inverse
[QUOTE=AlexP;705251] $\langle T(x),T(T^{-1}(x)) \rangle = \langle T^{-1}(T(x)),T^{-1}(T(T^{-1}(x))) \rangle$

I'm afraid this is not generally true.
It requires $T^{-1}$ to preserve the inner product, which is not automatically the case.