32.7. Diagonalization and Basis of Eigenvectors

Try these first

1. 1.

   What is diagonalization?

   Hint

   Think of transforming a matrix to make its operations simpler.

2. 2.

   Define an eigenvector.

   Hint

   Recall the relationship involving a scalar and a matrix.

3. 3.

   What is necessary for a matrix to be diagonalizable?

   • n distinct eigenvalues
   • n linearly independent eigenvectors
   • symmetric matrix

   Hint

   Think about the definitions of eigenvalues and eigenvectors.

4. 4.

   True or False: Diagonalization can simplify matrix calculations.

   • True
   • False

   Hint

   Remember the purpose of diagonalization.

5. 5.

   Here is a 2x2 matrix A: [[3, 1], [0, 2]]. Determine if it can be diagonalized and find the matrices P and D if it can.

   Hint

   Remember the properties of eigenvalues and eigenvectors.

6. 6.

   Consider the 2x2 matrix B = [[1, 1], [0, 1]]. Show that it cannot be diagonalized.

   Hint

   Evaluate eigenvalues and their counting carefully.