21.11. Diagonalization of Matrices

Try these first

1. 1.

   Define a diagonal matrix.

   Hint

   Think about the placement of zeroes in a matrix.

2. 2.

   What is an eigenvector?

   Hint

   Consider how vectors interact with linear transformations.

3. 3.

   What is a diagonal matrix?

   • A matrix with all zero elements
   • A matrix with non-zero values only on the diagonal
   • A square matrix with eigenvectors

   Hint

   Remember where zeroes are placed in such matrices.

4. 4.

   True or False: If a matrix has repeated eigenvalues, it is always diagonalizable.

   • True
   • False

   Hint

   Think about the conditions required for diagonalization.

5. 5.

   Given the matrix A = [[4, 1], [2, 3]], find the eigenvalues and determine whether A is diagonalizable. Justify your reasoning.

   Hint

   Use the characteristic polynomial to derive the eigenvalues.

6. 6.

   For the matrix B = [[2, -1], [0, 2]], analyze the multiplicities of eigenvalues and determine its diagonalizability.

   Hint

   Check the number of linearly independent eigenvectors compared to the eigenvalue's multiplicity.