In this paper, the authors elegantly present a LMI region as a subset $\mathcal{D}$ of the complex plane
$$ \mathcal{D} = \{z \in \mathbb{C} | L + z\cdot M + \bar{z}\cdot M^T < 0 \}$$ where $L = L^T$ and $M$ are some real matrices. Then, they proceed to say that a real matrix $A$ is $\mathcal{D}$ stable (that is, it has all eigenvalues in $\mathcal{D}$) iff there $\exists$ a symmetric positive definite matrix $X$, such that $$ L \otimes X + M \otimes (X\cdot A) + M^T \otimes (A^T\cdot X) < 0$$ where $\otimes$ denotes the Kronecker product. How can I prove this?
My ideea
Indeed, for $L = 0$ and $M = 1$ the LMI region is the left complex plane and indeed the above condition becomes the well known Lyapunov stability requirement for the dynamical system $\dot{x} = A\cdot x$ which is equivalent with the eigenvalues of $A$ being in the left complex plane! Am I suppose to see this as some sort of transformation of the imaginary axis in a convex region?