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?
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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.
Ruslan
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That leaves the RHS to zero, and then the solver returns a zero solution. – Antonio Mar 09 '17 at 17:04
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@Antonio right, and now I refer you to uniqueness theorem for Sylvester equation (see the link in my answer). – Ruslan Mar 09 '17 at 18:11