What does the eigenvalues tell you about the stability of a system?

Assume that you have a dynamical system that is represented by a matrix, where you can compute its eigenvalues and eigenvectors. How would these eigenvalues and eigenvectors would be different for stable vs. unstable systems? Does the existance of feedback loops have anything to do with it?

What does the eigenvalues tell you about the stability of a system?
In stable systems *all* the real part of all eigenvalues must be negative. If the complex parts exist the equilibrium takes the form of a focus, otherwise its called a knot. The eigenvectors determine the orientation of the equilibrium in phase space.





A feedback loop can be used to stabilize a system, but is in general non-linear. In that case the system can no longer be described by a matrix.
Reply:stable systems are the ones for negative eigenvalues


and unstabel systems are the ones for positive eigenvalues



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