Eigenvalue

Definition

An eigenvalue is a special scalar associated with a square matrix that indicates how much the corresponding eigenvector is stretched or compressed during a linear transformation. Eigenvalues help in understanding the behavior of matrices, especially in transformations, stability analysis, and systems of equations. They are crucial for determining properties like stability and oscillatory behavior in economic models.

5 Must Know Facts For Your Next Test

1. Eigenvalues can be real or complex numbers depending on the matrix's properties and its characteristics.
2. The sum of the eigenvalues of a matrix equals the trace of the matrix, which is the sum of its diagonal elements.
3. Eigenvalues can indicate stability: if all eigenvalues have negative real parts in a dynamic system, the system is stable.
4. To find eigenvalues, you solve the characteristic equation obtained from the determinant of \(A - \lambda I = 0\), where \(A\) is the matrix and \(I\) is the identity matrix.
5. In economics, eigenvalues are often used to analyze the stability of equilibrium points in dynamic models and systems.

Review Questions

• How do eigenvalues relate to eigenvectors in the context of linear transformations?
  • Eigenvalues and eigenvectors work together to describe how a matrix transforms vectors. When you apply a matrix to an eigenvector, it results in a new vector that points in the same direction but is scaled by the eigenvalue. This scaling shows how much the transformation stretches or compresses that vector. Thus, understanding both concepts helps analyze transformations and their effects on various systems.
• What role do eigenvalues play in determining the stability of economic models?
  • Eigenvalues are crucial for assessing stability in economic models, particularly in dynamic systems. When analyzing equilibrium points, if all eigenvalues have negative real parts, it indicates that small perturbations will decay over time, leading to stability. Conversely, if any eigenvalue has a positive real part, it suggests that perturbations will grow, leading to instability. This relationship makes eigenvalues essential tools for economists studying long-term behaviors of models.
• Evaluate how you could use eigenvalues to analyze a system's behavior over time and make predictions about future states.
  • To evaluate a system's behavior using eigenvalues, start by deriving them from the system's characteristic equation. By analyzing their values, you can determine if the system will stabilize or exhibit oscillatory behavior over time. For example, if all eigenvalues are negative, you can predict that any deviations from equilibrium will diminish. On the other hand, if there are positive eigenvalues, this might suggest potential growth or runaway effects. This predictive power allows economists and mathematicians to model future states effectively based on current conditions.

Related terms

Eigenvector:

An eigenvector is a non-zero vector that only changes by a scalar factor when a linear transformation is applied to it by a matrix.

Matrix: A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns, used to represent linear transformations.

Characteristic Polynomial:

The characteristic polynomial of a matrix is a polynomial whose roots are the eigenvalues of the matrix, typically derived from the determinant of the matrix subtracted by the scalar times the identity matrix.