Eigenvalues

Definition

Eigenvalues are special numbers associated with a square matrix that provide insight into the behavior of linear transformations represented by that matrix. Specifically, they represent the scaling factors by which eigenvectors are stretched or compressed during these transformations. Understanding eigenvalues is crucial in analyzing flow lines and equilibrium points, as they can indicate stability or instability in dynamic systems.

5 Must Know Facts For Your Next Test

1. Eigenvalues are found by solving the characteristic equation of a matrix, which is derived from the determinant equation |A - λI| = 0, where A is the matrix, λ represents the eigenvalue, and I is the identity matrix.
2. The number of eigenvalues for a given n x n matrix is equal to n, though some eigenvalues may be repeated or complex.
3. In the context of dynamical systems, a positive eigenvalue indicates instability, while a negative eigenvalue suggests stability at an equilibrium point.
4. If an eigenvalue is zero, it suggests that there is a direction in which the transformation represented by the matrix does not change the vector at all.
5. Eigenvalues can help determine the behavior of systems over time, including whether they converge to equilibrium points or diverge away from them.

Review Questions

• How do eigenvalues relate to stability in dynamical systems?
  • Eigenvalues provide critical information regarding stability in dynamical systems. If an equilibrium point has positive eigenvalues, small perturbations will cause the system to move away from equilibrium, indicating instability. Conversely, negative eigenvalues suggest that perturbations will decay over time, leading the system back to equilibrium and indicating stability. Therefore, analyzing eigenvalues helps predict how systems behave near equilibrium points.
• What role does the characteristic polynomial play in finding eigenvalues?
  • The characteristic polynomial is essential for determining eigenvalues from a matrix. It is formulated as the determinant equation |A - λI| = 0, where A is the matrix and λ represents potential eigenvalues. Solving this polynomial gives us the eigenvalues of the matrix. Each root corresponds to an eigenvalue, allowing for deeper insights into the transformation properties represented by A.
• Evaluate how understanding eigenvalues can impact our analysis of flow lines in a dynamic system.
  • Understanding eigenvalues significantly impacts our analysis of flow lines in dynamic systems by providing insights into how trajectories behave over time. Positive eigenvalues suggest that trajectories diverge from equilibrium points, indicating unstable flow lines. Negative eigenvalues show convergence towards equilibrium points, revealing stable flow lines. Thus, assessing eigenvalues allows us to predict how system states evolve and stabilize or destabilize over time.

Related terms

Eigenvectors: Vectors that only change by a scalar factor when a linear transformation is applied to them, corresponding to their associated eigenvalues.

Matrix: A rectangular array of numbers or functions arranged in rows and columns, used to represent linear transformations.

Characteristic Polynomial: A polynomial derived from a matrix that, when solved for its roots, yields the eigenvalues of that matrix.