21.11.2. Conditions for Diagonalizability

Try these first

1. 1.

   Define a diagonalizable matrix.

   Hint

   Remember the form of the diagonal matrix.

2. 2.

   What must a matrix have to be diagonalizable?

   Hint

   Think about the concepts of eigenvalues and eigenvectors.

3. 3.

   What is a necessary condition for a matrix to be diagonalizable?

   • It has at least one zero eigenvalue
   • It must have n linearly independent eigenvectors
   • It must be an identity matrix

   Hint

   Think about the definitions of eigenvectors.

4. 4.

   True or False: All matrices with distinct eigenvalues are diagonalizable.

   • True
   • False

   Hint

   Recall how eigenvalues relate to diagonalizability.

5. 5.

   Prove that the matrix A = [[1, 2], [0, 3]] is diagonalizable or not. Use computational methods to find eigenvalues and relations.

   Hint

   Calculate eigenvalues using the characteristic polynomial.

6. 6.

   Create a diagonalizable matrix of size 3x3 and show that it satisfies necessary diagonalizability conditions.

   Hint

   Choose distinct eigenvalues and construct accordingly.