I am looking to solve the following matrix equation for $G$


$$GHG + M = 0$$


where $G$, $H$, and $M$ are square, symmetric, real matrices. $H$ is negative-definite and $M$ is positive-definite. $G$ should also be positive-definite. 


Is it possible? 


Many thanks!



How to solve a non-homogeneous quadratic matrix equation?
(commenting about the equation with the plus sign, I hope that the correction was right).


This is one of the few quadratic matrix equations that have a closed form solution. Set $A=-H^{-1}$; then $G = A(A^{-1}M)^{1/2}=A^{1/2}(A^{-1/2}MA^{-1/2})^{1/2}A^{1/2}$. Here the operation $X^{1/2}$, for a matrix $X$, denotes the unique square root with spectrum in the right half-plane. The solution $G$ is known as the *matrix geometric mean* of the positive definite matrices $M$ and $A$, and denoted by $A\operatorname{\#}M$. You can see Section 4.1 of Bhatia's book *Positive definite matrices* for more information and proofs of these facts.


