\begin{equation} AX+XA=B \end{equation}
$A^{-1}$ and $B^{-1}$ is available.
The furthest I've gotten is: \begin{equation} X+A^{-1}XA=A^{-1}B \end{equation} which doesn't help at all.
2026-04-11 09:20:41.1775899241
Need a hint on how to solve this matrix equation
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This is called a Sylvester equation. Note that a unique solution will require that $A$ be non-singular and have no eigenvalue pairs of the form $\lambda$ and $-\lambda$. That Wikipedia page describes several well known solution methods that are implemented in standard mathematical packages.