The characteristic equation

The characteristic equation

Lessons

We say that a scalar λ\lambda is an eigenvalue of an n×nn \times n matrix AA if and only if λ\lambda satisfies the following characteristic equation:
det(AλI)=0\det(A-\lambda I)=0

We say that det(AλI)\det(A-\lambda I) is a characteristic polynomial.

Useful ways to find eigenvalues
When dealing with a 2×22 \times 2 matrix, use the formula det(A)=adbc\det (A) = ad-bc.

When dealing with a 3×33 \times 3 matrix, use the shortcut method.

When dealing with a triangular matrix, know that the determinant is just the product of the diagonal entries.

Note that:
1. An eigenvalue is a distinct root if it has a multiplicity of 1
2. An eigenvalue is a repeated root if it has a multiplicity greater than 1
  • 1.
    The Characteristic Equation Overview:
    a)
    What is the Characteristic Equation?
    det(AλI)=0\det (A-\lambda I)=0
    • The characteristic polynomial is det(AλI)\det (A-\lambda I)

    b)
    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

    c)
    Shortcut to Determinants of Matrices
    2×22 \times 2 matrices
    3×33 \times 3 matrices
    • Triangular matrices

    d)
    Eigenvalue with Multiplicity
    • Distinct eigenvalues
    • Repeated eigenvalues


  • 2.
    Finding the Characteristic Polynomial
    Find the characteristic polynomial of AA if:
    Find the characteristic polynomial

  • 3.
    Find the characteristic polynomial of AA if:
    Find the characteristic polynomial

  • 4.
    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.

  • 5.
    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.

  • 6.
    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.

  • 7.
    Proofs dealing with the Characteristic Equation
    Show that AA and ATA^T has the same characteristic polynomials.

  • 8.
    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