Suppose I have a symmetric matrix $S$ (unknown) to be computed, which is represented by: $$ S = \sum_{i=0}^{\infty} A^{i\top}A^{i} = I + A^{\top}A + A^{2\top}A^{2} + \cdots, $$ where powers of $A$ (known) would decay to 0.
Now I want a decomposition of $S$, such that: $$S = R^{\top}R.$$
Since $S$ is unknown, how can I get this decomposition $R^{\top}R$? Is there any relationship between $R$ and $A$, thus I don't need to compute $S$ to decompose $S$.
To follow up on a comment by user1551, we have a relationship $$S-A^TSA = I$$ which admits a closed-form solution for $S$. $$\eqalign{ (I\otimes I - A^T\otimes A^T)\,{\rm vec}(S) = {\rm vec}(I) \cr {\rm vec}(S) = (I\otimes I - A^T\otimes A^T)^{-1}\,{\rm vec}(I) \cr S = {\rm devec}\Big((I\otimes I - A^T\otimes A^T)^{-1}\,{\rm vec}(I)\Big) \cr }$$ Now choose a factorization (Cholesky) that has the desired form.