# Analytic solution to a nested discrete lyapunov equation?

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!

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