I was recently made aware in another post of the Lyapunov equation, so now I am trying to understand it a bit better and therefore have two questions.
In the following, we assume that $M \in \mathbb{R}^{n \times n}$ such that all eigenvalues of $M$ have real parts and $\Sigma \in \mathbb{R}^{n \times n}, P \in \mathbb{R}^{n \times n}$ will always be symmetric matrizes.
Now, the Lypanuov equation in this situation reads $$ M\Sigma + \Sigma M^T = P,$$ and the associated theory assures that for $P$ positive definite, there is always a $\Sigma$ positive definite such that the above equation is fullfilled.
Now, my first question would be, given fixed $M$ and $\Sigma$ positive definite, can we somehow find a simple criterion whether the result is positive definite? That is, can we easily read out whether a $\Sigma$ solves a Lyapunov equation for some $P$?
My second question would now be, what happens if we have that $\Sigma$ is only positive semi definite?
It seems like in this situation, one can still tweak around to get a positive semi-definite $P$: Assume we know the Kernel $K$ of $\Sigma$, and define an operator $U$ that is $\mu \ Id$ on $K$ for $\mu>=0$ and $0$ else (this should be possible to be taken symmetric by taking using the decomposition $\Sigma = ODO^T$ and replacing the diagonals of $D$ appropiately). Then $$ \Sigma + U$$ is a positive definite matrix.
Thus, if we could show that $\Sigma + U$ solves a Lyapunov equation for $P$ then $\Sigma$ would also solve a Lyapunov equation for the positive semidefinite matrix $P- (MU + UM^T)$, where we have to impose some conditions on $\mu$. (If I am not mistaken, $\mu< \frac{p_{\min}}{2|m_{max}|}$ should be enough, where $p_{\min}$ is the smallest eigenvalue of $P$ and $m_{\max}$ the largest one of $M$). Does this construction work indeed or did I make a mistake somewhere?