Self-referencing Lyapunov equation

Question

Let $A$ be a square, Hurwitz matrix. The Lyapunov-like equation

$$ AP + PA^T = -P $$

is trivially solved by $P=0$, where $0$ here has the same size of $A$.

Can we find positive-definite solutions too?

Answer 1

HINT:

Move $-P$ to the LHS, then introduce identity matrix factor $I$ to collect the $P$ factors with either $A$ or $A^T$. Then you'll get a Sylvester equation without any "self-references". Using uniqueness theorem for it (see the link), you can check whether the solution you find is the unique solution.