Hello members, let's consider the following equation
$$X=F\_{1}XF\_{1}^{T}+...+F\_{p}XF\_{p}^{T}+C$$
where $p$ is an positive integer and $C$ is a known positive semidefinite matrix. If we augment $F=[F\_{1}...F\_{p}]$ and $Y=diag (X...X)$, then the equation becomes
$$FYF^{T}−[I...0]Y[I...0]^{T}+C=0$$
seems like a generalized Lyapunov equation. However, there is a constraint on $Y$ for its diagonal form. How to compute $X$?


I met with this problem for dealing with stability analysis for dynamic systems with multiple multiplicative noise.



On solution of a class of discrete-time  Lyapunov equation 
A trivial way (not necessarily the one you'd use, unless the matrices were very large) to solve this problem is to rewrite it using Kronecker product notation. We use the observation


\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\*}


You can try solving this linear system using any standard method, taking care to implement the matrix-vector product to take advantage of the Kronecker product structure. 


