33.1. Diagonalization of a Matrix

Try these first

1. 1.

   Define a diagonal matrix.

   Hint

   Think about where non-zero elements can be placed in the matrix.

2. 2.

   What is an eigenvalue?

   Hint

   Consider what happens to an eigenvector when a linear transformation is applied.

3. 3.

   What is a diagonal matrix?

   • A matrix with all elements zero
   • Non-zero elements only along the diagonal
   • All non-diagonal elements are non-zero

   Hint

   Think about where you can find numbers in a diagonal matrix.

4. 4.

   True or False: A matrix with repeated eigenvalues is always diagonalizable.

   • True
   • False

   Hint

   Consider what happens when there are fewer eigenvectors than algebraic multiplicities.

5. 5.

   Given the matrix A = [3, 2; 4, 5], diagonalize it if possible. Show all steps.

   Hint

   Start with finding the characteristic polynomial.

6. 6.

   Determine whether the matrix B = [1, 0; 0, 1] can be diagonalized. Explain why or why not.

   Hint

   What do you notice about the form of this matrix?