Suppose we have a positive definite matrix $Q \in \mathbb{R}^{n\times n}$, are there general conditions on $A \in \mathbb{R}^{m\times n}$ such that $A Q^{-1}A^T$ is positive semi-definite ($A Q^{-1}A^T \succeq 0$)? Or positive definite ($A Q^{-1}A^T \succ 0$)?
Extension: Suppose there are no general conditions on $A$ such that $A Q^{-1}A^T$ can be made positive semi-definite or positive definite given that $Q$ is positive definite. By further assuming that $Q$ is symmetric, i.e., $Q = Q^T$, are there any conditions on $A$ such that $A Q^{-1}A^T$ can be made either positive semi-definite or positive definite?
If $Q$ is positive semidefinite, then $AQ^{-1}A^T$ will always be positive semidefinite.
If $Q$ is positive definite, then it can be written in the form $$ Q = H + K, $$ where $H = \frac 12(Q + Q^T)$ is symmetric and positive definite and $K = \frac 12(Q - Q^T)$ is skew-symmetric. Because $H$ is positive definite, there exists an invertible matrix $P$ such that $S = PP^T$, and we have $$ P^{-1}QP^{-T} = P^{-1}HP^{-T} + P^{-1}KP^{-T} = I + P^{-1}KP^{-T} $$ where $I$ denotes the identity matrix. Note in particular that $J:= P^{-1}KP^{-T}$ is a skew-symmetric matrix. Note that $Q^{-1}$ is positive definite if and only if the matrix $$ P^TQ^{-1}P = [P^{-1}QP^{-T}]^{-1} = [I + J]^{-1} $$ is positive definite. Thus, we have reduced the question to that of whether/when $[I + J]^{-1}$ is positive definite for a skew-symmetric $J$.
Note that a matrix $M$ is positive definite if and only if the symmetric matrix $M + M^T$ is positive semidefinite. Taking $M = [I + J]^{-1}$, we have $$ M + M^T = [I + J]^{-1} + [I + J]^{-T} = [I + J]^{-1} + [I - J]^{-1}. $$ I claim that this can be rewritten as $$ [I + J]^{-1} + [I - J]^{-1} = 2[(I + J)(I - J)]^{-1} = 2 (I - J^2)^{-1}. $$ Now, we note that $I - J^2$ is necessarily positive definite because it is the sum of a positive definite matrix $I$ and a positive semidefinite matrix $-J^2 = JJ^T$. Thus, $(I - J^2)^{-1}$ and hence $(I + J)^{-1}$ is positive definite.