# 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

## -Free-

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