Why is this matrix product diagonalizable?
Since $A$ is positive-definite, there exists an invertible square root of the matrix which is also symmetric. Denote this as $A^\frac{1}{2}$. Then $$A^{-\frac{1}{2}}ABA^\frac{1}{2} = A^\frac{1}{2}BA^\frac{1}{2}$$ where the latter is symmetric because $B$ and $A^\frac{1}{2}$ are both symmetric. Therefore $AB$ is similar to a symmetric matrix and hence diagonalizable.