Given is a symmetric and positive-definite solution $C$ to the discrete lyapunov equation,
$WCW^T = C - M$
where $M$ is again symmetric and positive-definite. Is there an analytic solution to
$WXW^T = X - C$
in terms of $C$? $W$ is big ($n > 1000$) and sparse.
Of course you can vectorize the equation using Kronecker products or write the solution as an infinite sum over $W^k CW^{kT}$ but I have the feeling there is a much simpler and elegant solution to it.
Thanks!