Eigenvalue and eigenvectors

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Intros
Lessons
  1. Eigenvalues and Eigenvectors Overview:
  2. Definition of Eigenvalues and Eigenvectors
    • What are eigenvectors?
    • What are eigenvalues?
  3. 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
  4. Eigenspace
    • What is an eigenspace?
    • Finding a basis for the eigenspace
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Examples
Lessons
  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?
    1. Let Verifying Eigenvectors. Is is this vector an eigenvector 3 an eigenvector of AA? If so, find the eigenvalue.
      1. 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.
        1. Finding a Basis for the Eigenspace
          Find a basis for the corresponding eigenspace if:
          Finding a Basis for the Eigenspace
          1. Proof Related to Eigenvalues and Eigenvectors
            Prove that if A2A^2 is the zero matrix, then the only eigenvalue of AA is 0.
            1. Let λ\lambda be an eigenvalue of an invertible matrix AA. Then λ1\lambda ^{-1} is an eigenvalue of A1A^{-1}.