eigenvalues and eigenvectors

(2.1 hours to learn)


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.


This concept has the prerequisites:


  • 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)


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


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


Khan Academy: Linear Algebra

See also