33.4. Procedure to Diagonalize a Matrix

Try these first

1. 1.

   What is the characteristic polynomial of the matrix A = [[1, 2], [3, 4]]?

   Hint

   Use the formula det(A - λI) = 0.

2. 2.

   Find the eigenvalue of the matrix A = [[5, 0], [0, 5]].

   Hint

   This is a diagonal matrix, so the eigenvalues are on the diagonal.

3. 3.

   What must be true for a matrix to be diagonalizable?

   • It has n linearly independent eigenvectors.
   • It must быть symmetric.
   • It contains only positive eigenvalues.

   Hint

   Consider the definitions related to diagonalization criteria.

4. 4.

   True or False: The eigenvalues must all be distinct for a matrix to be diagonalizable.

   • True
   • False

   Hint

   Think about the conditions required for diagonalization.

5. 5.

   Given the matrix A = [[1, 2], [2, 1]], determine whether it is diagonalizable. Calculate its eigenvalues and eigenvectors step by step.

   Hint

   Ensure you correctly compute the determinant and solve the eigenvalue equation.

6. 6.

   Consider the matrix B = [[1, 1], [0, 1]]. Prove that it cannot be diagonalized, discussing the implications of repeated eigenvalues.

   Hint

   Reflect on the relationship between eigenvalues and the linear independence of eigenvectors.