4.2 Finding Eigenvalues and Eigenvectors

Reading

Complete the preview activity and read  Chapter 4 Section 2 in Understanding Linear Algebra by David Austin.

Do you remember?

Given a matrix [latex]A[/latex], a non-zero vector [latex]\vec{v}[/latex] is an eigenvector of [latex]A[/latex] if [latex]A\vec{v}[/latex] is a multiple of [latex]\vec{v}[/latex]. In other words, [latex]\vec{v}[/latex] is an eigenvector of A if [latex]A\vec{v}= \lambda\vec{v}[/latex]. Note that [latex]\lambda[/latex] can equal zero, but that the zero vector is not considered an eigenvector.

 

Finding eigenvalues, derivation of the method

Unfortunate typo: video should say “finding eigenvalues”

Examples of finding eigenvalues:

 

The polynomial [latex]P(\lambda) = det(A-\lambda I)[/latex] is called the characteristic polynomial. If A is an nxn matrix, this polynomial has degree n. Its roots are the eigenvalues of A.

Finding eigenvectors, derivation of the method

Examples of finding eigenvectors:

The eigenvectors of a matrix A for a given eigenvalue [latex]\lambda[/latex] form a subspace (we also throw in [latex]\vec{0}[/latex]), called the eigenspace of A for eigenvalue [latex]\lambda[/latex]. This is written [latex]E_{\lambda} = Nul(A-\lambda I)[/latex].

 

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Math 220, Matrices Copyright © 2018 by Kristen Pueschel is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.

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