Could anyone help me solve this equation? Let $n,m,l$ be positive integers such that $n \ge m$ and $l>m$, but there is no such constraint between $l$ and $n$.
We have two known matrices $Q_1 \in \mathbb{R}^{n \times m}$ and $Q_2 \in \mathbb{R}^{n \times l}$.
The unknowns are $S \in \mathbb{R}^{m \times m}$ symmetric, $B \in \mathbb{R}^{m \times l}$ and $C \in \mathbb{R}^{l \times l}$ symmetric. The equations are $$\begin{align} S^2 + B C B^T &= Q_1^T Q_1 \\ B^T B + C &= Q_2^T Q_2 \\ S B + B C &= Q_1^T Q_2 \end{align} $$
One can write it in a structured way $$ \begin{pmatrix} S & B \\ B^T & I \end{pmatrix} \begin{pmatrix} I & 0 \\ 0 & C \end{pmatrix} \begin{pmatrix} S & B \\ B^T & I \end{pmatrix} = \begin{pmatrix} Q_1^T \\ Q_2^T \end{pmatrix} \begin{pmatrix} Q_1 & Q_2 \end{pmatrix} $$ I have a suspicion that this is a type of Ricatti equation. If it helps I have come to this set of equations by trying to solve the larger system $$ \begin{align} P_1 S + P_2 B^T &= Q_1\\ P_1 B + P_2 &= Q_2\\ P_1^T P_1 &= I\\ P_1^T P_2 &= 0, \end{align} $$ where $P_1$, $P_2$ are additional unknowns, which have the same size as $Q_1$, $Q_2$, respectively.