3
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ This equation looks like a special case of the sylvester equation. Maybe the transformations on that are helpful. $\endgroup$ – Bort Apr 30 '13 at 8:57
  • 1
    $\begingroup$ 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 Jun 15 '13 at 22:01

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.