Lyapunov equation

Hello Inzaghina! If examples are all your looking for then the most trivial and easiest to find are to either set $A = I$ , the Identity matrix, or $X = I$ . In the first case then you have that

$2X = C$

so choose any arbitrary X matrix and the $C$ is given by the above. In the second case you have that

$A + A^T = C$

so this time pick any matrix for A and then use the above to give you an example of $C$ .

I have met this beast before but have to say that in my experience it's usually the case that you know $A$ and $C$ and need to find $X$ !