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 for systems with multiplicaitve noise
linear algebra;matrices;matrix equations