Matrix equation with projection matrix

Question

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?

Answer 1

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 $$G_1G_2P^\top(PAP^\top)^{-1}P=G_1 G_2(G_1G_2A^{-1})^\top\bigl(G_1G_2(G_1G_2A^{-1})^\top\bigr)^{-1}G_1G_2A^{-1}=G_1G_2A^{-1}.$$