The characteristic equation

Lessons

• Introduction
  The Characteristic Equation Overview:
  
  a)

  What is the Characteristic Equation?
  • $\det (A-\lambda I)=0$
  • The characteristic polynomial is $\det (A-\lambda I)$
  
  
  b)

  Finding the Eigenvalue
  • The eigenvalues of a $2 \times 2$ matrix
  • The eigenvalues of a $3 \times 3$ matrix
  • The eigenvalues of a triangular matrix
  
  
  c)

  Shortcut to Determinants of Matrices
  • $2 \times 2$ matrices
  • $3 \times 3$ matrices
  • Triangular matrices
  
  
  d)

  Eigenvalue with Multiplicity
  • Distinct eigenvalues
  • Repeated eigenvalues
  
  

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• 1.
  Finding the Characteristic Polynomial
  Find the characteristic polynomial of $A$ if:
  

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• 2.
  Find the characteristic polynomial of $A$ if:
  

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• 3.
  Finding the Eigenvalues of a $2 \times 2$ matrix
  Find all the eigenvalues of the matrix
  State their multiplicities, and what type of eigenvalues they are.

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• 4.
  Finding the Eigenvalues of a $3 \times 3$ matrix
  Find all the eigenvalues of the matrix
  State their multiplicities, and what type of eigenvalues they are.

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• 5.
  Finding the Eigenvalues of a triangular matrix
  Find all the eigenvalues of the matrix
  State their multiplicities, and what type of eigenvalues they are.

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• 6.
  Proofs dealing with the Characteristic Equation
  Show that $A$ and $A^T$ has the same characteristic polynomials.

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• 7.
  Suppose $A$ is an $n \times n$ triangular matrix where all the diagonal entries are $c$. Then the characteristic polynomial is
  $p(\lambda)=(c-\lambda)^n$

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