Given $A,B$ symmetric positive definite matrices, is $B^{-1}A$ always symmetric positive definite?

Question

Given $A$ and $B$ to be symmetric positive definite matrices. Is the product $B^{-1}A$ always positive definite and symmetric?

Answer 1

No for symmetry. Symmetry is not true because:

$(B^{-1} A)^T = A^T B^{{-1}^T} = A B^{T^{-1}} = A B^{-1}$

The above only equals the original equation if $A$ and $B^{-1}$ are commutative, which isn't an inherent property of positive definite matrices. The only case I know of in which they are commutative is when they can both be diagonalized and they have the same eigenvector basis. Only the first one is guaranteed for these matrices as symmetric implies diagonalizable.

Further evidence: https://math.vanderbilt.edu/sapirmv/msapir/jan22.html#:~:text=If%20the%20product%20of%20two,BA%20then%20AB%20is%20symmetric.

A question that should address the eigenvalue part: Is the product of symmetric positive semidefinite matrices positive definite?