I hope you're having a wonderful day. I'm stuck trying to solve this exercise from 'Linear Algebra' by author Marco Manetti. It's marked as a hard one, so I'm not surprised by its difficulty. However, since I have spent hours on it at this point, I would really like some closure and thus would appreciate if you could help me with hints or a solution.
The exercise is the following: "Suppose $A$ and $B$ are symmetric, positive semi-definite real matrices. Prove that $AB + I$ is invertible and that the eingevalues of $AB$ are nonnegative."
I have a proof of the fact that: "If $B$ is a real $n$ by $n$ matrix $B^TB + aI$ is invertible $\forall a \in \mathbb{R}$."
Thus, my line of thinking has been the following: If I manage to write $AB$ as $C^TC$ for some matrix $C$, $AB + I$ is certainly invertible. Moreover, if $\lambda$ is negative real number $p_{AB}(\lambda) = \det(AB + (-\lambda)I) \neq 0$ as the argument isn't invertible because of the lemma. Therefore, $\lambda$ can't be an eigenvalue. However, my whole argument crumbles apart because I don't know whether $C$ can even be proved to exist and, if that's the case, I don't know how to find it.
Any help is thoroughly appreciated.