Is there any general method to solve a matrix equation which may have some following form: (with $X$ is what needed to find)($A,B,C,D,X$ are all $n \times n$ matrices)(and in my particular case, $n=3$)
$AX+XB+XCX+D=0$
$X^T+AX^{-1}B=0$
$A+X^TBX=0$
If there is no general method, what are some kinds of equation considered classical and solvable.
Note: They are separate equations.
Thanks very much! These kinds of equations appear in my research in physics.
Well, you have 3 equations and 5 unknowns. With algebraic manipulations (by substituting for $A$ and then for $B$) you can get it to the form $$-X^TX + I + XC + D = 0.$$
This way you're left with one equation and three unknowns. I don't think you can take it much further. The point is that you cannot solve for $X$ uniquely.