How to solve this quadratic matrix equation?

Question

I have to solve the following equation in matrix $D$

$$D^{T} D(D V D^{T} + I)^{-1} = A$$

where $I$ is the identity matrix, are $V$ and $A$ are given matrices. Does anyone have any idea how to solve it for $D$, please? Thanks.

Answer 1

The equation in $D$, $D^TD=ADVD^T+A$ is difficult to study. More precisely, I think that there does not exist any method that permits to solve the previous equation in the generic case (when we randomly choose the $(n\times n)$ real matrices $A,V$).

For example, when $n=2$, we obtain (generically) $8$ solutions and, when $n=3$, $112$ solutions (over the complex numbers, $D\in M_n(\mathbb{C})$).