I am trying to solve the equation $AX + XA' = 0$ I could find how to solve when "$+$" is a "$-$" or $X$ is conjugated instead of $A$. Is there a solution for this problem too?
In particular, I am looking for a solution where all of the matrices are $\mathbb{C}^{n\times n}$ and $A=BC$ where both B and C are hermitian symmetric and positive definite.