[Math] Products of positive definite matrices

Question

Having two positive definite matrices $A, B$, it holds that the product $ABA$ is positive definite.

I'm looking for a simple proof of this fact.

Answer 1

You don't even need positive definiteness of $A$, it is enough that $B$ is positive definite and $A$ is nonsingular and symmetric. Then for any nonzero column $x \in \mathbb{R}^n$ column $Ax$ is also nonzero, therefore $$ x^T(ABA)x = (Ax)^T B (Ax) > 0. $$