33.7. Non-Diagonalizable Matrices and Jordan Form (Brief Note)

Try these first

1. 1.

   What makes a matrix non-diagonalizable?

   Hint

   Consider the relationship between eigenvalues and eigenvectors.

2. 2.

   Describe how Jordan form can be useful for studying non-diagonalizable matrices.

   Hint

   Think about how block structures can help with computations.

3. 3.

   What is the Jordan form used for?

   • Diagonalizing matrices
   • An alternative for non-diagonalizable matrices
   • Simplifying symmetric matrices

   Hint

   Think about what you do when diagonalization fails.

4. 4.

   True or False: All matrices in civil engineering are diagonalizable.

   • True
   • False

   Hint

   Consider the definitions we've discussed.

5. 5.

   For the matrix A = [1 1; 0 1], determine if it is diagonalizable. If not, describe its Jordan form.

   Hint

   Check the algebraic and geometric multiplicities and how they relate.

6. 6.

   Find a non-diagonalizable matrix of size 3 and explain its Jordan form.

   Hint

   Consider how repeated eigenvalues lead to Jordan blocks.