32.5. Example

Try these first

1. 1.

   What is the characteristic equation of the matrix A = [[4, 1], [0, 4]]?

   Hint

   Remember, I is the identity matrix.

2. 2.

   What is an eigenvalue?

   Hint

   Think about how eigenvalues relate to transformations.

3. 3.

   What does the characteristic equation tell us?

   • Determines eigenvalues
   • Finds determinants
   • Calculates traces

   Hint

   Focus on how the term 'characteristic' relates to eigenvalues.

4. 4.

   True or False: The eigenspace dimension is always equal to the algebraic multiplicity of the eigenvalue.

   • True
   • False

   Hint

   Review the definitions of AM and GM.

5. 5.

   Given a new matrix B = [[4, 2], [0, 4]], calculate its eigenvalues and explain if it is diagonalizable.

   Hint

   Check the relationship between AM and GM.

6. 6.

   Suppose a matrix has real eigenvalues and the calculation reveals three linearly independent eigenvectors. What can be said about its diagonalizability?

   Hint

   Recall the definition of diagonalizability.