Suppose I have a matrix $M$, such that:
$$M = \sum_{k=0}^{\infty} c^{k}(A^{k})^{\top}DA^{k} = D + cA^{\top}DA + c^{2}A^{2\top}DA^{2} + \cdots,$$ where $c \in [0,1]$ is a scalar, D is a diagonal matrix with $(1-c)<D_{ii}<1$, $A$ is a column stochastic matrix (sum of elements of each column is 1). (This guarantee that this series would converge).
How can I get a facorization(decomposition) of M by $A$, such that:
$$M = R^{\top}R.$$
where $R$ is some matrix. We know that such a decomposition must exist, since $M$ is symmetric.
The problem is that I want this decomposition is expressed by $A$, i.e., $R = f(A)$, $f$ is some function work on $A$.
How can I get this decomposition (factorization) of $M$?
Thanks.