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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.

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  • $\begingroup$ This equation looks like a special case of the sylvester equation. Maybe the transformations on that are helpful. $\endgroup$
    – Bort
    Commented Apr 30, 2013 at 8:57
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    $\begingroup$ 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. $\endgroup$
    – Federico Poloni
    Commented Jun 15, 2013 at 22:01

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