Eigenvalues

Definition

Eigenvalues are special numbers associated with a square matrix that provide important information about the linear transformations represented by that matrix. When a matrix acts on a vector, the eigenvalues tell us how much the eigenvector is stretched or shrunk and in which direction it is pointing. These concepts are foundational in understanding vector spaces, linear transformations, and dynamic systems, allowing us to analyze stability and behavior over time.

5 Must Know Facts For Your Next Test

1. Eigenvalues can be real or complex numbers, depending on the properties of the matrix.
2. The calculation of eigenvalues involves solving the characteristic polynomial, which is obtained from the determinant of \(A - \lambda I\), where \(A\) is the matrix, \(\lambda\) represents the eigenvalue, and \(I\) is the identity matrix.
3. In dynamic systems, eigenvalues can indicate stability; if all eigenvalues have negative real parts, the system tends to be stable.
4. The geometric multiplicity of an eigenvalue refers to the number of linearly independent eigenvectors associated with it, which helps understand the transformation's behavior.
5. Eigenvalues are crucial in various applications, including differential equations, optimization problems, and modeling economic systems to predict long-term behavior.

Review Questions

• How do eigenvalues relate to linear transformations and what do they reveal about the transformation's effect on vectors?
  • Eigenvalues help us understand how a linear transformation affects vectors by showing how much an eigenvector is stretched or compressed during the transformation. If an eigenvalue is greater than one, it indicates that the corresponding eigenvector gets stretched; if it's less than one but positive, it gets compressed. If the eigenvalue is negative, it implies a reflection along with stretching or compressing. This relationship highlights how eigenvalues provide insight into the nature of the transformation represented by a matrix.
• Discuss how you would find eigenvalues for a given matrix and why this process is important for analyzing stability in dynamic systems.
  • To find the eigenvalues of a matrix, you start by computing the characteristic polynomial by calculating the determinant of \(A - \lambda I\) and setting it equal to zero. This equation provides the eigenvalues as its roots. Understanding these eigenvalues is crucial for analyzing stability in dynamic systems because they inform us whether perturbations will grow or diminish over time. Specifically, if all eigenvalues have negative real parts, any small disturbance will decay back to equilibrium, indicating stability.
• Evaluate the implications of having complex eigenvalues in a system of differential equations and how this affects system behavior.
  • Complex eigenvalues in a system of differential equations indicate oscillatory behavior within the system. When an eigenvalue has both real and imaginary parts, it suggests that solutions will exhibit sinusoidal patterns while growing or decaying based on the real part. This characteristic is crucial in understanding not just stability but also how oscillations evolve over time within economic models or physical systems. The interplay between these complex values can lead to insights about cycles and periodic behaviors that are significant in various fields.

Related terms

Eigenvectors:

Vectors that, when transformed by a matrix, only change in scale and not in direction. Each eigenvector corresponds to its associated eigenvalue.

Determinant:

A scalar value that is a function of the entries of a square matrix, which helps to determine whether the matrix is invertible and also plays a role in finding eigenvalues.

Characteristic Polynomial:

A polynomial whose roots are the eigenvalues of a matrix. It is derived from the determinant of the matrix subtracted by a variable times the identity matrix.