Condition on output of Lyapunov equation being positive definite

Question

I have Matlab code that solves the Lyapunov equation $$AX + XA^T + Q = 0$$ for a 3-D array of matrices, $A$, using the Matlab function lyap(A,Q). My problem is that sometimes the resultant matrix, $X$, is positive definite and sometimes it is not. My question is given that the matrix $Q$ is positive definite, what is the condition on $A$ that the resultant matrix, $X$, is always positive definite?

Answer 1

Given $Q>0$, $X$ is positive definite if and only if $A$ is stable (i.e., all the eigenvalues of $A$ have negative real parts).

This is the Lyapunov Theorem. See thm 2.2.1 in the book "Topics in Matrix Analysis".

In fact, the above condition may still be true even if $Q$ is merely positive semi-definite. But in that case $(A,Q)$ must be controllable (If $Q$ is positive definite then $(A,Q)$ is controllable). For proof see thm 2.4.7 in the book. You can also have a look at this post.