Suppose there are three matrices $A,B,C\in \mathbb{R}^{n\times n}$, full rank. Suppose any entries of A is different from zero. Suppose $A$ and $C$ to be positive definite and symmetric.
Suppose the matrix $C$ is the following product: $C=BAB'$
Moreover, $C$ can be partitioned as follows: (for some $m<n$)
$C=\begin{pmatrix}C_{11} & 0_{m\times m} \\ 0_{(n-m)\times(n-m)} & C_{22} \end{pmatrix}$
that is, for some $m$, the off-diagonal (wrt the partition) blocks are null, and $C_{11}$ is a $(n-m)\times m$ matrix, $C_{22}$ is a $m \times (n-m)$.
Can we retrieve any info or infer any (non-trivial and sharp) restrictions on the matrix $B$?
[through any decomposition or treating $B$ as unknown,...] Thank you in advance!