31.4. Diagonalization and Similarity

Try these first

1. 1.

   Define what it means for a matrix to be diagonalizable.

   Hint

   Think about what forms a matrix can take when diagonalized.

2. 2.

   What is an eigenvector?

   Hint

   Consider how these vectors behave during transformation.

3. 3.

   What does it mean for a matrix to be diagonalizable?

   • It can be transformed into a lower triangular matrix.
   • It can be represented as D = P^{-1}AP.
   • It has no eigenvalues.

   Hint

   Focus on how D changes the representation of A.

4. 4.

   True or False: A matrix with multiple identical eigenvalues cannot be diagonalized.

   • True
   • False

   Hint

   Consider the importance of eigenvector independence.

5. 5.

   Let A = [[2, 1], [1, 2]]. Find the eigenvalues and demonstrate whether A is diagonalizable.

   Hint

   Calculate the characteristic polynomial to find the eigenvalues.

6. 6.

   Consider a symmetric matrix A. Prove that it has real eigenvalues and therefore is diagonalizable.

   Hint

   Refer to the properties of symmetric matrices and the spectral theorem.