# 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:

- matrix multiplication (Eigenvectors and eigenvalues can be used to analyze repeated matrix multiplications.)
- roots of polynomials (The eigenvalues can be computed as roots of a polynomial.)
- linear systems as matrices (Eigenvalues are defined as solutions to a system of linear equations.)
- complex numbers (The eigenvalues and eigenvectors may be complex valued.)
- determinant (The characteristic polynomial, used for computing eigenvalues analytically, is given in terms of a determinant.)
- column space and nullspace (Eigenspaces can be viewed as nullspaces of a particular matrix.)

## 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.

Additional dependencies:

- complex vectors and matrices

→ MIT Open Courseware: Linear Algebra (2011)

Videos for an introductory linear algebra course focusing on numerical methods.

Location:
Lecture "Eigenvalues and eigenvectors"

## -Paid-

→ Introduction to Linear Algebra

An introductory linear algebra textbook with an emphasis on numerical methods.

Location:
Section 6.1, "Introduction to eigenvalues," pages 283-291

→ Multivariable Mathematics

A textbook on linear algebra and multivariable calculus with proofs.

Location:
Section 9.2, "Eigenvalues, eigenvectors, and applications," up to "Diagonalizability," pages 422-429

Additional dependencies:

- change of basis

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

## -Free-

→ Khan Academy: Linear Algebra

## See also

- Eigenvalues are closely related to the characteristic polynomial of a matrix.
- The determinant of a matrix is the product of the eigenvalues .
- The Spectral Theorem states that symmetric matrices have a full set of eigenvalues with orthogonal eigenvectors.
- Eigenvalues are used in solving linear ordinary differential equations .