33.8. Diagonalization of Symmetric Matrices

Try these first

1. 1.

   Define a symmetric matrix and provide an example.

   Hint

   Think about the definition of the matrix being equal to its transpose.

2. 2.

   What is the significance of real eigenvalues in symmetric matrices?

   Hint

   Consider how this affects calculations.

3. 3.

   What characterizes a symmetric matrix?

   • A = A^T
   • A = -A
   • A is always diagonal

   Hint

   Recall the definition of symmetric matrices.

4. 4.

   True or False: All eigenvectors of a symmetric matrix are not necessarily orthogonal.

   • True
   • False

   Hint

   Consider eigenvector properties we just discussed.

5. 5.

   Consider the symmetric matrix A = [2 1; 1 2]. Diagonalize the matrix and explain the significance of each eigenvalue.

   Hint

   Calculate eigenvalues and eigenvectors using the characteristic polynomial.

6. 6.

   Show that the matrix A = [1 2; 2 1] can be expressed as A = QDQ^T. Determine Q and D.

   Hint

   Work through the steps of finding characteristic polynomial and eigenvalues first.