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I know that the matrix equation $A X + X A' + C = 0$ is in the form of the time-continuous Lyapunov equation, so solving for $X$ is pretty trivial since the solution already and numeric solvers already exists. However, I don't know how to solve for $X$ when $B X B'$ is added to the equation so that the equation reads:

$$A X + X A' + B X B' + C = 0$$

This is no longer in a form I am familiar with and I was wondering if there is any known solution out there. It should be noted that both $A$ and $B$ are Hermitian ($A= A'$ and $B= B'$) for my purposes. If it helps, $C$ is an identity matrix for my purposes as well.

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  • $\begingroup$ There are various norms about whether or not to use TeX when plain text can do, but, if you do use plain text, then please be careful about tripping the Markdown parser: AX + XA' A*X + X*A' is parsed with "X + X" in italics, which is not what you want. $A X + X A'$ A X + X A' is almost certainly preferable. I have edited accordingly. $\endgroup$
    – LSpice
    Commented Mar 9, 2021 at 19:58
  • $\begingroup$ Although similar to the continuous Linear Quadratic Ricatti Equation, to consult here, it is not. May I ask where does this equation comes from? $\endgroup$
    – Bruno Lobo
    Commented Mar 9, 2021 at 22:33
  • $\begingroup$ @BrunoHenriquePeixoto yeah, so I do a lot of research into open quantum system using density matrix formalism and the differential equation describing the system of spins at steady state basically boils down to this equation. I can solve this as a function of time easily, but computationally it’s very slow and I need something much faster for my purposes. $\endgroup$
    – Elias Frantz
    Commented Mar 10, 2021 at 0:58
  • $\begingroup$ Are you sure the statement leads to it? Take a look on the Ricatti equation formalism as I mentioned. What is the mathematical statement? $\endgroup$
    – Bruno Lobo
    Commented Mar 11, 2021 at 1:12
  • $\begingroup$ I recommend to: guess an initial solution for X e.g. the identity matrix, set the RHS as the equation error, calculate its norm and iterate the norm with the algorithm github.com/brunolnetto/baryopt for some high iterations number. $\endgroup$
    – Bruno Lobo
    Commented Mar 11, 2021 at 1:45

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