This is a follow up question on from How to solve a non-homogeneous quadratic matrix equation?.

Given the matrix $G = A(A^{-1}M)^{1/2}=A^{1/2}(A^{-1/2}MA^{-1/2})^{1/2}A^{1/2}$, where $A=-H^{-1}$, with $H$ symmetric negative definite and $M$ symmetric positive definite.

Is it possible to establish a clear relationship between the eigenvectors and eigenvalues of $G$, and those of $H$ and $M$?

Also, I have read that $G$ is on the midpoint of the geodesic linking $A$ and $M$, in the space of symmetric positive definite matrices, with distance between two matrices $A$ and $M$ defined as $$ d(A,M) = \| \log(A^{-1/2}MA^{-1/2}) \|_{\mathrm{HS}},$$ where $ \| X \|_{\mathrm{HS}} = \mathrm{Trace} (X^TX)^{1/2}$ . Do you know if it would be possible to have a 3D graphic representation of this relationship for $2\times 2$ matrices, and if so, how one would go about it?

Once again, many thanks.