The following matrices are all $n\times n$ and real.
$B$ and $D$ are unknown. Moreover, $D$ is diagonal, and $diag(D)=diag(B)$.
$\Sigma_{11}$, $\Sigma_{21}$ and $\Sigma_{22}$ are all known. $\Sigma_{11}$ and $\Sigma_{22}$ are positive definite (if it matters).
These matrices satisfy the following equation:
$\Sigma_{11} + \Sigma_{21}D' + B\Sigma_{21}' + B\Sigma_{22}D' = 0$
The unknown matrices have $n^2$ independent entries ($B$ has $n^2$ unknown entries, and the $n$ entries of $D$ are found on the diagonal of $B$). So I suspect that this has a (probably non-unique) solution.
I can solve for $B$ in the scalar case where $n=1$, but am stumped when $n>1$. Any ideas? If necessary, we can assume $B$, $D$, or $\Sigma_{21}$ are invertible, but I would prefer not to.
Thanks!