I am trying to solve the next problem \begin{aligned} & \underset{P}{\text{maximize}} & & \log \det P \\ & \text{subject to} & & A^T P^{-1} + P^{-1} A \preceq 0 \\ &&& P^{-1} \succeq 0 \\ &&& \ldots \end{aligned}
Is this problem convex or can be transfrom to convex?
I try to introduce the new variable Q
\begin{aligned} & \underset{P}{\text{maximize}} & & \log \det P \\ & \text{subject to} & & A^T Q + P Q \preceq 0 \\ &&& Q \succeq 0 \\ &&& P Q = I &&& \ldots \end{aligned} but last constrainst is BMI.