91Ó°ÊÓ

In the study of linear algebra and dynamic systems, eigenvalues play a crucial role. They are numbers associated with a square matrix that provide valuable insights about the behavior of linear transformations represented by matrices. An eigenvalue is determined by solving the equation \( A\vec{v} = \lambda\vec{v} \), where \( A \) is a matrix, \( \lambda \) is an eigenvalue, and \( \vec{v} \) is the corresponding eigenvector.
In the context of our system, eigenvalues help us understand the long-term behavior of the system. Specifically, they determine whether the system will converge to an equilibrium point or diverge away. This makes eigenvalues fundamental in determining the stability of dynamic systems.
When examining the matrix associated with a dynamic system, solving for eigenvalues is one of the first steps to analyze stability.

The characteristic equation is a polynomial whose roots are the eigenvalues of a matrix. For a given square matrix \( A \), the characteristic equation is obtained by calculating the determinant of \( A - \lambda I \) and setting it to zero, resulting in \( \det(A - \lambda I) = 0 \).
For our specific example, the matrix \( A \) is:

• \( \begin{bmatrix} 2 & -4 \ 5 & -6 \end{bmatrix} \)

Add \( \lambda I \) (where \( I \) is the identity matrix) and compute the determinant to come up with the characteristic polynomial \( \lambda^2 + 4\lambda + 8 = 0 \).
The characteristic equation is essential because it directly provides us with the eigenvalues of the matrix, which we need to determine the system's stability.

When solving the characteristic equation, sometimes we encounter complex numbers as eigenvalues. These arise particularly when the discriminant of the polynomial is negative, meaning the polynomial has no real roots. Instead, the solutions are complex conjugates.
For the system matrix \( A \), we found a characteristic equation \( \lambda^2 + 4\lambda + 8 = 0 \) with a negative discriminant \(-16\). This results in complex eigenvalues \( \lambda = -2 \pm 2i \).
Complex eigenvalues often indicate oscillatory behavior in systems. However, the real part of these eigenvalues influences stability. With a negative real part in our eigenvalues, the system shows damped oscillation, suggesting stable equilibrium.

In dynamic systems, stability of equilibrium refers to whether a system will return to an equilibrium state after a slight disturbance. This is crucial for determining the system's behavior over time.
For our example, we analyze the system's stability by looking at the eigenvalues of the matrix \( A \). We found that the eigenvalues are complex with a negative real part: \( \lambda = -2 \pm 2i \). This indicates that any small perturbations will diminish over time, leading the system back to equilibrium.
Stability is determined primarily by the sign of the real parts of the eigenvalues:

• Negative real parts mean the system returns to equilibrium (stable).
• Positive real parts suggest the system will move away from equilibrium (unstable).
• Zero real parts require further investigation, as they could indicate neutral stability or a more complicated behavior.

In conclusion, complex eigenvalues with negative real parts assure us that our equilibrium point is stable.