Eigenvalue

Definition

An eigenvalue is a scalar value that describes how a linear transformation affects a vector in a vector space. Specifically, when a linear transformation represented by a matrix is applied to an eigenvector, the result is the same as multiplying that eigenvector by the eigenvalue. This concept is crucial in various mathematical fields, including linear algebra and quantum mechanics, where it helps relate transformations to physical properties and measurable quantities.

5 Must Know Facts For Your Next Test

1. Eigenvalues can be found by solving the characteristic polynomial of a matrix, which is obtained from the equation det(A - λI) = 0, where A is the matrix, λ represents the eigenvalue, and I is the identity matrix.
2. In quantum mechanics, eigenvalues correspond to observable quantities such as energy levels or momentum, linking mathematical theory with physical measurements.
3. The number of eigenvalues of an n x n matrix is always equal to n, although some eigenvalues may be repeated or complex.
4. Eigenvalues can be real or complex numbers, which provides insight into the stability and behavior of systems described by matrices.
5. The sum of the eigenvalues of a matrix equals its trace, which is the sum of its diagonal elements.

Review Questions

• How does the concept of eigenvalues connect to physical properties measured in quantum mechanics?
  • Eigenvalues are fundamental in quantum mechanics as they correspond to measurable quantities like energy levels. When operators associated with physical observables act on state vectors (eigenvectors), the results yield these eigenvalues, indicating potential outcomes for measurements. This relationship showcases how abstract mathematical concepts directly relate to tangible physical phenomena in quantum systems.
• Discuss how you would find the eigenvalues of a given matrix and what significance they hold in linear transformations.
  • To find the eigenvalues of a matrix, you set up the characteristic polynomial by calculating det(A - λI) = 0. Solving this polynomial gives you the eigenvalues. These values reveal how certain vectors (eigenvectors) are scaled under the transformation represented by the matrix. They help us understand properties like stability and dynamics in systems modeled by these matrices.
• Evaluate the implications of having complex eigenvalues for a system represented by a specific matrix and how this affects its behavior over time.
  • Complex eigenvalues suggest oscillatory behavior in systems modeled by matrices. When you have complex conjugate pairs of eigenvalues, it indicates that solutions may involve exponential growth or decay combined with rotations or oscillations. This behavior has significant implications in stability analysis; for instance, if any part of the spectrum has positive real parts, it could signal instability in dynamic systems over time.

Related terms

Eigenvector:

An eigenvector is a non-zero vector that, when a linear transformation is applied, changes only in scale and not in direction.

Matrix: A matrix is a rectangular array of numbers or functions arranged in rows and columns, which can represent linear transformations.

Linear Transformation:

A linear transformation is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication.