33.5. Example

Try these first

1. 1.

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

   Hint

   Use det(A - λI) to derive it.

2. 2.

   Define what eigenvalues are in relation to a matrix.

   Hint

   Think about the effect of a transformation on vectors.

3. 3.

   What is the significance of the characteristic polynomial in diagonalization?

   • It provides eigenvalues
   • It gives eigenvectors
   • It's irrelevant

   Hint

   Think about where eigenvalues come from.

4. 4.

   True or False: Every matrix can be diagonalized.

   • True
   • False

   Hint

   Consider the conditions for diagonalizability.

5. 5.

   Given the matrix A = [[5, 1], [2, 4]], find its eigenvalues and demonstrate the diagonalization process.

   Hint

   Follow the characteristic polynomial and eigenvector process.

6. 6.

   Consider the matrix A = [[3, 2], [4, 1]]. Is it diagonalizable? Justify your answer with calculations.

   Hint

   Always compare the algebraic and geometric multiplicities.