# On solution of a class of discrete-time Lyapunov equation for systems with multiplicaitve noise

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.

• This equation looks like a special case of the sylvester equation. Maybe the transformations on that are helpful. – Bort Apr 30 '13 at 8:57
• Your best shot is probably vectorizing everything and solving the resulting $n^2\times n^2$ linear system. As far as I know, there is no $O(n^3)$ direct solver for dense matrix equations of the form $\sum_i A_i X B_i = C$ with more than 2 terms. It must be another instance of the "2 is easy, 3 is hard" meta-phenomenon that we encounter in several branches of mathematics. – Federico Poloni Jun 15 '13 at 22:01