I want to find an expression for the inverse of the matrix system $Z=CAC^T$, where $A \in \mathbb{C}^{n \times n}$ is block diagonal with dense blocks, and $C \in \{-1,0,1\}$ with dimension $m \times n$, and $m < n$. The context of this problem is that this matrix represents a physical system partitioned into volumes, with boundary conditions being imposed by the use of a constraint matrix $C$.
So far I have tried two approaches. My first attempt involved writing out the simplest possible scenario:
$(C_1 ~~ C_2)\begin{pmatrix}A_1 & \\ & A_2\end{pmatrix}\begin{pmatrix}C_1^T\\C_2^T\end{pmatrix}x = \left(C_1 A_1 C_1^T + C_2 A_2 C_2^T\right)x$.
Unfortunately, this takes us right back to where we started, except now we have two smaller problems of the same form.
My second attempt was to extract an expression from the second term on the right-hand side of the identity $(A+UV^T)^{-1}=A^{-1}-A^{-1}U(I+V^TA^{-1}U)^{-1}V^TA^{-1}$, taken with $U=V=C^T$. This yielded the following expression:
$CAC^T = I-(CC^T)\left\lbrace CA^{-1}(I-(A^{-1}+C^TC)^{-1}C^T\right\rbrace^{-1}(CC^T)$.
Gross. Does anyone have any ideas as to how to proceed in perhaps simplifying this, or a suggestion for a different approach? I can't imagine this problem is uncommon. I have done many Google/MathExchange searches for terms like "matrix quadratic form inverse" but nothing for this particular problem has surfaced. A similar question was asked last year, but my problem has different properties in that my middle matrix (here, $A$) is not s.p.d., nor is it generally symmetric.