Eigenvalue and eigenvectors

Eigenvalue and eigenvectors

Lessons

An eigenvector of an n×nn \times n matrix AA is a non-zero vector xx such that Ax=λxAx= \lambda x, for some scalar λ\lambda. The scalar λ\lambda is called the eigenvalue.

We say the eigenvector xx corresponds to the eigenvalue λ\lambda.

Given an eigenvalue λ\lambda of matrix AA, we can find a corresponding eigenvector xx by solving
(AλI)x=0(A-\lambda I)x=0
And finding a non-trivial solution xx.

The eigenspace is the null space of the matrix AλIA-\lambda I. In other words, the eigenspace is a set of all solutions for the equation
(AλI)x=0(A-\lambda I)x=0

Of course, we can find the basis for the eigenspace by finding the basis of the null space of AλIA-\lambda I.
  • Introduction
    Eigenvalues and Eigenvectors Overview:
    a)
    Definition of Eigenvalues and Eigenvectors
    • What are eigenvectors?
    • What are eigenvalues?

    b)
    Verifying Eigenvalues/Eigenvectors of a Matrix
    • Eigenvectors: Show that Ax=λxAx=\lambda x
    • Eigenvalues: Get a non-trivial solution for (AλI)x=0(A-\lambda I)x=0
    • Finding a eigenvector given an eigenvalue

    c)
    Eigenspace
    • What is an eigenspace?
    • Finding a basis for the eigenspace


  • 1.
    Verifying Eigenvectors
    Let Verifying Eigenvectors. Is is this vector an eigenvector 1 an eigenvector of AA? If so, find the eigenvalue. What about is this vector an eigenvector 2?

  • 2.
    Let Verifying Eigenvectors. Is is this vector an eigenvector 3 an eigenvector of AA? If so, find the eigenvalue.

  • 3.
    Verifying Eigenvalues and finding a corresponding eigenvector
    Let Verifying Eigenvalues and finding a corresponding eigenvector. Is λ=1\lambda=1 an eigenvalue of AA? If so, find a corresponding eigenvector.

  • 4.
    Finding a Basis for the Eigenspace
    Find a basis for the corresponding eigenspace if:
    Finding a Basis for the Eigenspace

  • 5.
    Proof Related to Eigenvalues and Eigenvectors
    Prove that if A2A^2 is the zero matrix, then the only eigenvalue of AA is 0.

  • 6.
    Let λ\lambda be an eigenvalue of an invertible matrix AA. Then λ1\lambda ^{-1} is an eigenvalue of A1A^{-1}.