Product of positive definite and negative definite commuting matrices Suppose I have two real-valued symmetric matrices $A, B$.Let $A$ be positive-definite and $B$ be negative-definite, and also let $AB = BA$. 
Consider now the product, $C = AB$, which is guaranteed to be real since $A$ and $B$ are real, and it is also guaranteed to be symmetric since $[A,B] = 0$. Is $C$ negative-definite? Is there a simple proof of this fact? 

Proof 1. Note that $A$ and $B$ are simultaneously unitarily diagonalisable.
Proof 2. Note that $A^{1/2}$ is a polynomial in $A$. Hence it commutes with $B$ and $AB=A^{1/2}BA^{1/2}$.
