29.8. Example Problems

Try these first

1. 1.

   What is the characteristic equation of a 2x2 matrix?

   Hint

   Think of how you find eigenvalues.

2. 2.

   Define eigenvalue in your own words.

   Hint

   Consider stretching and direction.

3. 3.

   What do we call the equation used to find eigenvalues?

   • Eigenvalue Equation
   • Characteristic Equation
   • Determinant Equation

   Hint

   It's the equation that helps in finding eigenvalues.

4. 4.

   Eigenvectors are always parallel to the original vectors.

   • True
   • False

   Hint

   Consider eigenvector definitions.

5. 5.

   Given a 3x3 matrix A = [[2, 1, 1], [1, 2, 1], [1, 1, 2]], calculate its eigenvalues and eigenvectors.

   Hint

   Use the characteristic polynomial for step one.

6. 6.

   Why is it important for the eigenvector matrices corresponding to distinct eigenvalues to be orthogonal in real symmetric matrices?

   Hint

   Consider how these directions are used in practical applications.