The characteristic equation

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Intros
Lessons
  1. The Characteristic Equation Overview:
  2. What is the Characteristic Equation?
    det(AλI)=0\det (A-\lambda I)=0
    • The characteristic polynomial is det(AλI)\det (A-\lambda I)
  3. Finding the Eigenvalue
    • The eigenvalues of a 2×22 \times 2 matrix
    • The eigenvalues of a 3×33 \times 3 matrix
    • The eigenvalues of a triangular matrix
  4. Shortcut to Determinants of Matrices
    2×22 \times 2 matrices
    3×33 \times 3 matrices
    • Triangular matrices
  5. Eigenvalue with Multiplicity
    • Distinct eigenvalues
    • Repeated eigenvalues
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Examples
Lessons
  1. Finding the Characteristic Polynomial
    Find the characteristic polynomial of AA if:
    Find the characteristic polynomial
    1. Find the characteristic polynomial of AA if:
      Find the characteristic polynomial
      1. Finding the Eigenvalues of a 2×22 \times 2 matrix
        Find all the eigenvalues of the matrix
        Find all the eigenvalues of the 2 x 2 matrix
        State their multiplicities, and what type of eigenvalues they are.
        1. Finding the Eigenvalues of a 3×33 \times 3 matrix
          Find all the eigenvalues of the matrix
          Find all the eigenvalues of the 3 x 3 matrix
          State their multiplicities, and what type of eigenvalues they are.
          1. Finding the Eigenvalues of a triangular matrix
            Find all the eigenvalues of the matrix
            Find all the eigenvalues of the triangular matrix
            State their multiplicities, and what type of eigenvalues they are.
            1. Proofs dealing with the Characteristic Equation
              Show that AA and ATA^T has the same characteristic polynomials.
              1. Suppose AA is an n×nn \times n triangular matrix where all the diagonal entries are cc. Then the characteristic polynomial is
                p(λ)=(cλ)np(\lambda)=(c-\lambda)^n