33.3. Diagonalization Criteria

Try these first

1. 1.

   What is an eigenvalue?

   Hint

   Think about how eigenvalues relate to the actions on eigenvectors.

2. 2.

   How do you determine if a matrix is diagonalizable?

   Hint

   Remember what you learned about linear independence.

3. 3.

   A matrix is diagonalizable if it has:

   • At least one eigenvalue
   • n linearly independent eigenvectors
   • Only distinct eigenvalues

   Hint

   Focus on the number of eigenvectors needed.

4. 4.

   True or False: A matrix can be diagonalized if its eigenvalue has an algebraic multiplicity greater than its geometric multiplicity.

   • True
   • False

   Hint

   Think about the definitions of both multiplicities.

5. 5.

   Prove that the matrix [[2, 0], [0, 2]] is diagonalizable.

   Hint

   Calculate the eigenvalues first and verify independence.

6. 6.

   Given the matrix [[0, 1], [0, 0]], discuss whether it can be diagonalized and justify.

   Hint

   Remember to check the eigenvectors corresponding to the eigenvalue.