The direction field shown in the image suggests a saddle point, where the trajectories are moving away along one direction and towards the equilibrium point along another. This behavior is characteristic of a system with eigenvalues of opposite signs: one positive (indicating instability in one direction) and one negative (indicating stability in the perpendicular direction).

Given this interpretation, the correct answer is:

• The eigenvalues have opposite sign.

Let me know if you want further details or have any questions.

Related Questions

1. What does the presence of a saddle point imply about the stability of the system?
2. How can we determine the stability of a system based on eigenvalues alone?
3. What other types of equilibrium points can a linear system have?
4. How do eigenvalues affect the direction of flow in a direction field?
5. What happens when both eigenvalues are positive or negative in a linear system?

Tip: For linear systems, a saddle point is indicative of mixed stability: instability in one direction and stability in another.