Linear Algebra Difficulty: Intermediate

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors characterize the invariant directions of linear transformations. They are essential for stability analysis, dimensionality reduction, and graph methods.

Key Points

  • An eigenvector $v$ satisfies $Av = \lambda v$.
  • The characteristic polynomial $\det(A - \lambda I) = 0$ gives eigenvalues.
  • Spectral theorem: symmetric matrices have real eigenvalues and orthogonal eigenvectors.

Formulas

Eigenvalue equation
$$A\mathbf{v} = \lambda \mathbf{v}$$
Characteristic polynomial
$$p(\lambda) = \det(A - \lambda I)$$
Spectral theorem (symmetric)
$$A = A^\top \implies A = Q\Lambda Q^\top$$

Code Example

import numpy as np

A = np.array([[4, 2], [1, 3]])
eigenvalues, eigenvectors = np.linalg.eig(A)
print(eigenvalues)
print(eigenvectors)

Tags

  • spectral-theory
  • stability

References

  • Linear Algebra Done Right
    Sheldon Axler · Springer · source
  • Introduction to Linear Algebra
    Gilbert Strang · Wellesley-Cambridge Press · source

Knowledge Graph