33.13. Practice Problems

Try these first

1. 1.

   Diagonalize the matrix A = [6 -2; 2 2].

   Hint

   Calculate the eigenvalues first.

2. 2.

   Find the eigenvalues of the matrix A = [1 1; 0 1].

   Hint

   Use the characteristic polynomial.

3. 3.

   What is the form of a diagonalized matrix?

   • PDP-1
   • DP-P1
   • PDP^-1

   Hint

   Remember the relationship between the matrices.

4. 4.

   True or False: All symmetric matrices are diagonalizable.

   • True
   • False

   Hint

   Consider the properties of symmetric matrices.

5. 5.

   Given a matrix A = [3 1; 0 3], prove its non-diagonalizability.

   Hint

   Examine the eigenspace to conclude.

6. 6.

   A symmetric matrix B = [2 1; 1 2] is given, find its eigenvalues and eigenvectors analytically.

   Hint

   Use the characteristic polynomial to derive eigenvalues.