The following matrices are all $n\times n$ and real.
$A$, $B$, $C$, and $D$ are unknown. Moreover, $D$ is diagonal, and $diag(D)=diag(B)$. $A$ and $C$ are both positive definite.
$\Sigma_{11}$, $\Sigma_{21}$ and $\Sigma_{22}$ are all known. $\Sigma_{11}$ and $\Sigma_{22}$ are positive definite.
These matrices satisfy the following:
$\Sigma_{11} = BAB' + DCD'$
$\Sigma_{21} = BA + DC $
$\Sigma_{22} = A + C$
The unknown matrices have $2n^2 + n$ independent entries ($B$ has $n^2$, $A$ and $C$ each have $\frac{n^2+n}{2}$, and the $n$ entries of $D$ are found on the diagonal of $B$). Because of the symmetric structures, the matrix equations are really $\frac{n(n+1)}{2} + n^2 +\frac{n(n+1)}{2} = 2n^2 + n$ scalar equations, so I suspect that this has a solution. (I also suspect that the solution is unique up to signs/reordering for the $n>1$ case.)
I am interested in solving for $B$ ($A$ and $C$ are unimportant). I can do so in the scalar case where $n=1$, but am stumped when $n>1$. Any ideas? If necessary, we can assume $B$ or $\Sigma_{21}$ are invertible, but I would prefer not to.
Thanks!