eigenvalues and eigenvectors

(2.1 hours to learn)

Summary

If A is a square matrix, the eigenvalues are the scalar values u satisfying Ax = ux, and the eigenvectors are the values of x. Eigenvectors and eigenvalues give a convenient representation of matrices for computing powers of matrices and for solving differential equations. An important special case is the spectral decomposition of symmetric matrices.

Context

This concept has the prerequisites:

Goals

  • Know the definitions of eigenvalues and eigenvectors
  • Understand why the eigenvectors for a given eigenvalue form a subspace
  • Be able to calculate eigenvalues and eigenvectors in terms of the roots of the characteristic polynomial
  • Show that the sum of the eigenvalues equals the trace
  • Show that the product of the eigenvalues equals the determinant
  • Know why eigenvalues and eigenvectors can sometimes be complex valued
  • Show that the complex eigenvalues of a real matrix come in conjugate pairs
  • Show that eigenvectors corresponding to distinct eigenvalues are linearly independent

Core resources (read/watch one of the following)

-Free-

A First Course in Linear Algebra (2012)
A linear algebra textbook with proofs.
Author: Robert A. Beezer
Additional dependencies:
  • complex vectors and matrices
MIT Open Courseware: Linear Algebra (2011)
Videos for an introductory linear algebra course focusing on numerical methods.
Author: Gilbert Strang

-Paid-

Supplemental resources (the following are optional, but you may find them useful)

-Free-

Khan Academy: Linear Algebra

See also