Eigenvalues

Definition

Eigenvalues are special scalar values associated with a square matrix that provide important information about the behavior of linear transformations. They represent the factors by which corresponding eigenvectors are stretched or compressed during transformation, and play a crucial role in understanding system dynamics, stability, and behavior over time.

5 Must Know Facts For Your Next Test

1. Eigenvalues can be real or complex numbers and are calculated by solving the characteristic polynomial obtained from the determinant of the matrix minus lambda times the identity matrix.
2. In phase portraits for linear systems, the sign and magnitude of eigenvalues determine the type of equilibrium point and its stability properties.
3. For discrete systems, eigenvalues can indicate whether iterations converge or diverge, thus affecting long-term system behavior.
4. In the analysis of periodic orbits, eigenvalues can be used to assess the stability of the orbit; if they lie inside the unit circle, the orbit is stable.
5. Delay differential equations often involve eigenvalues to understand stability and determine how delays influence system dynamics.

Review Questions

• How do eigenvalues influence the stability analysis of linear systems?
  • Eigenvalues are crucial for determining the stability of linear systems. The sign of the real parts of eigenvalues helps classify equilibrium points; if all eigenvalues have negative real parts, the system is stable and returns to equilibrium after perturbation. Conversely, if any eigenvalue has a positive real part, it indicates instability as trajectories move away from equilibrium. Thus, analyzing eigenvalues provides insights into how systems react over time.
• Discuss how the concept of eigenvalues applies in the context of stability analysis for discrete systems.
  • In discrete systems, eigenvalues determine whether sequences converge to a fixed point or diverge. If all eigenvalues lie within the unit circle in the complex plane, then iterative sequences will converge towards an equilibrium point, indicating stability. However, if any eigenvalue lies outside this circle, it signals divergence and instability. This relationship is key in predicting long-term behavior in dynamical systems.
• Evaluate how changes in eigenvalues can affect the stability of periodic orbits in dynamical systems.
  • Changes in eigenvalues can significantly impact the stability of periodic orbits by altering their behavior under perturbations. If the eigenvalues of a linearized system around a periodic orbit shift into or out of the unit circle as parameters change, it indicates a bifurcation event where the stability shifts from stable to unstable or vice versa. This sensitivity to eigenvalue changes highlights the importance of numerical bifurcation analysis in understanding how slight modifications in system parameters can lead to dramatic changes in dynamics.

Related terms

Eigenvectors: Vectors that do not change direction when a linear transformation is applied, only scaled by their corresponding eigenvalue.

Characteristic Polynomial: A polynomial equation derived from a square matrix that is used to find its eigenvalues.

Stability: The property of a dynamical system to return to equilibrium after a disturbance, often analyzed using eigenvalues.