32.12. Diagonalizability and Basis of Eigenvectors

Try these first

1. 1.

   Define diagonalizability in your own words.

   Hint

   Consider the form A = PDP⁻¹.

2. 2.

   What is an eigenvalue?

   Hint

   Think about how it relates to eigenvectors in the equation Av = λv.

3. 3.

   What is the main condition for a matrix to be diagonalizable?

   • It has at least one eigenvalue.
   • It has n linearly independent eigenvectors.
   • It has distinct eigenvalues.
   • None of the above.

   Hint

   Think about how many vectors are needed to span the space.

4. 4.

   True or False: All symmetric matrices are diagonalizable.

   • True
   • False

   Hint

   Recall the properties of symmetric matrices.

5. 5.

   Given a 3x3 matrix with eigenvalues 2 (multiplicity 2) and 3 (multiplicity 1), explain whether it can be diagonalized and justify your answer.

   Hint

   Explore geometric vs algebraic multiplicity.

6. 6.

   Demonstrate the process of diagonalization for a symmetric matrix with eigenvalues 4, 5, and 6. Calculate P and D, showing all steps.

   Hint

   Use the characteristics of symmetric matrices.