The following matrices are all $n\times n$ and real.
$\beta$, $\Sigma_{11}$, $\Sigma_{21}$ and $\Sigma_{22}$ are all known. Moreover, $\left(\begin{array}{cc} \Sigma_{11} & \Sigma_{21}' \\ \Sigma_{21} & \Sigma_{22} \end{array}\right)$ is positive definite.
$A$, $H$, $D_u$, and $D_v$ are unknown. Moreover, $D_u$ and $D_v$ are diagonal.
These matrices satisfy the following:
$\Sigma_{11} = AD_vA' + \beta AD_uA'\beta'$
$\Sigma_{21} = HD_vA' + AD_uA'\beta'$
$\Sigma_{22} = HD_vH' + AD_uA'$
$D_v + D_u = I$
The unknown matrices have $2n^2 +2n$ independent entries. Because of the symmetric and diagonal structures, the matrix equations are really $\frac{n(n+1)}{2} + n^2 +\frac{n(n+1)}{2} + n = 2n^2 +2n$ scalar equations. So I suspect that this has a solution, unique up to signs/reordering. I can solve the scalar case where $n=1$, but am stumped when $n>1$. Any ideas? If necessary, we can assume $\beta$ or $\Sigma_{21}$ are invertible, but I would prefer not to.
Thanks!