I am encountering the following set of three matrix equations for which I search a solution in terms of ${\bf M}\in\mathbb{R}^{N\times N}$ and ${\bf D}\in\mathbb{R}^{Q\times N}$,
$${\bf M}{\bf W} = \gamma{\bf M},$$
$${\bf D}{\bf W} = \gamma{\bf D},$$
and $\bf W$ is given by self-consistency equation
$${\bf W} = {\bf D}^\top{\bf D} + {\bf M}^\top{\bf W}^{-1}{\bf M}.$$
This set of equation looks innocent enough, but the self-consistency gets me completely stuck. Any ideas or hints on how to solve such a system (if possible)?
Assume that $\gamma\in \mathbb{R}$.
The last relation implies $W^2=D^TDW+M^TW^{-1}MW=\gamma(D^TD+M^TW^{-1}M)=\gamma W$. Since $W$ is invertible, $W=\gamma I_N$ where $\gamma\not=0$.
It remains the equation $\gamma I_N=D^TD+1/\gamma M^TM$, that is, $M^TM=\gamma^2I_N-\gamma D^TD$.
We choose a matrix $D$ with non-zero singular values $(\sigma_i)_{i\leq k}$. Then $M$ exists iff $\gamma^2I_N-\gamma D^TD$ is symmetric $\geq 0$, that is, iff for every $i\leq k$, $\gamma^2-\gamma{\sigma_i}^2\geq 0$.