I'm currently working on a multidisciplinary research project about the structural controllability of brain networks. Specifically, I have constructed the adjacency matrix of brain networks and studied their controllability given a different input matrix $B$. However, I've received a comment about my work that I'm not sure how to approach.
The question is: "Is the $A$ matrix that is estimated stable, i.e., are the real parts of all the eigenvalues negative? If not, then the model in equation 1 is describing the structural brain as an unstable dynamic system."
I've considered the dynamics in brain networks by linear models, and equation 1 in the comment points out $\dot{x}(t) = Ax(t) + Bu(t)$. My constructed adjacency matrices are zero-one $90 \times 90$ and symmetric. Therefore, as far as I know, their eigenvalues are real. But, when I calculated the eigenvalues, I found that it has a combination of negative and positive values, and the number of positive values is almost thirty out of ninety.
I would like to know if my idea is appropriate for this situation. From my understanding, if $\dot{x} = Ax$ for a system, and all the eigenvalues of $A$ are negative, then the system is considered stable. Conversely, if all the eigenvalues are positive, then the system is considered unstable. However, when we encounter a system with some positive eigenvalues in controllability, we can drive the system into stability by properly selecting the inputs $Bu$, $\dot{x} = Ax + Bu$.
I would appreciate any help in dealing with this comment regarding my work.