We shall reconsider the following equation $$X=F_{1}XF_{1}^{T}+...+F_{p}XF_{p}^{T}+C$$ where $p$ is a positive integer and $C$ is a known symmetric positive semidefinite matrix.

I met with this problem for dealing with stability analysis for filtering problems with multiple multiplicative noise.

Recall the following property of the Kronecker product \begin{equation*} \text{vec}(AXB) = (B^T \otimes A)\text{vec}(X), \end{equation*} where the $\text{vec}(\cdot)$ stacks columns of $X$ into one long vector.

Defining now $x := \text{vec}(X)$ and $c = \text{vec}(C)$, with the above observation the original equation can be written as the following large (but highly structured) linear system:

\begin{equation*} \left(\sum\nolimits_i F_i \otimes F_i - (I \otimes I)\right)x = c. \end{equation*}

In a special case, for $i=1$, the equation reduces to the normal discrete-time Lyapunov equation, and if $F$ is stable, the solution will be symmetric positive semidefinite, i.e. $$X=F_{1}XF_{1}^{T}+C$$ with $X=\sum\nolimits_{j}^{\infty}F_{1}^{j}C(F_{1}^{T})^{j}$

My question is that, given a symmetric positive semidefinite matrix $C$, under what conditions can we find a symmetric positive semidefinite solution $X$?

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