Consider a matrix that satisfies
$$ \bf M = {\begin{bmatrix} \bf A ^\top & \bf B ^\top \end{bmatrix}} {\begin{bmatrix} \bf A \bf A ^\top + \bf C \bf C ^\top & \bf A \bf B ^\top \\ \bf B \bf A ^\top & \bf B \bf B ^\top \end{bmatrix}}^{-1} {\begin{bmatrix} \bf A \\ \bf B \end{bmatrix}} $$
where $\bf A$, $\bf B$, and $\bf C$ are matrices as well. I am attempting to calculate
$$\lim_{n \to \infty} {\bf{M}}^n$$
Notice that if ${\bf C} = {\bf 0}$ this would be trivial, since then ${\bf{M}}^n={\bf{M}}$ for all $n$. I have tried using the block matrix inversion formula to explicitly invert the middle matrix but I was not able to obtain the result that way. Any suggestion would be much appreciated.
Using the block-matrix inversion formula and simplifying, it follows that
$$ \bf M = \bf E+(E-I)D(E-I), $$
where
\begin{align} \bf D \equiv & \; \bf A^{\top}(A(I-E)A^{\top}+BB^{\top})^{-1}A,\\ \bf E \equiv & \; \bf C^{\top}(CC^{\top})^{-1}C. \end{align}
Notice that $\bf E$ is idempotent. Therefore,
$$ {\bf M}^n = {\bf E+(E-I)(D(E-I))}^n.$$
Hence, for instance, if the eigenvalues of $\bf D(E-I)$ all have modulus less than $1$, then $\lim_{n\to \infty}{\bf M}^n=\bf E$. In general
$$ \lim_{n\to \infty}{\bf M}^n = {\bf E +(E-I)}\lim_{n\to \infty}{\bf (D(E-I))}^n.$$