I need to solve the following equation for $P \in\mathbb{R}^{r\times d}$


$$P - G\_1G\_2(\lambda P^\top(PAP^\top)^{-1}P + A^{-1} ) = 0,$$


where the other quantities are known: $A\in\mathbb{R}^{d\times d}$, $G\_1 \in\mathbb{R}^{r\times d}$, $G\_{2} \in\mathbb{R}^{d\times d}$, $\lambda\in\mathbb{R}$.


I have already tried everything I know. If the equation had involved scalar quantities the solution would simply be
$$
p = g\_1g\_2(a^{-1}(1+\lambda)),
$$


but unfortunately in the matrix case I don't know how to proceed. Maybe there should be some way to decompose it using SVD or some trace trick?



Matrix equation with projection matrix
The solution for $P$ to
$$P - G\_1G\_2(\lambda P^\top(PAP^\top)^{-1}P + A^{-1} ) = 0$$
is
$$P=(1 +\lambda )G\_1 G\_2 A^{-1},$$
as one can check by substitution into
$$G\_1G\_2P^\top(PAP^\top)^{-1}P=G\_1 G\_2(G\_1G\_2A^{-1})^\top\bigl(G\_1G\_2(G\_1G\_2A^{-1})^\top\bigr)^{-1}G\_1G\_2A^{-1}=G\_1G\_2A^{-1}.$$


