28.13. Diagonalization of Linear Transformations

Try these first

1. 1.

   What does it mean for a matrix to be diagonalizable?

   Hint

   Think of the format of the matrix expression.

2. 2.

   What is the condition for a matrix to be diagonalizable?

   Hint

   Consider what makes eigenvalues distinct.

3. 3.

   What is a diagonalizable matrix?

   • A matrix that can be inverted
   • A matrix that can be expressed as A = PDP⁻¹
   • A non-square matrix

   Hint

   Consider the definition you learned.

4. 4.

   True or False: A matrix with repeated eigenvalues can automatically be considered diagonalizable.

   • True
   • False

   Hint

   Think about what you learned regarding conditions for diagonalizability.

5. 5.

   Given a matrix A with eigenvalues 2, 1, and 1, analyze if it is diagonalizable and justify your reasoning in two steps.

   Hint

   Review how to compute eigenvectors.

6. 6.

   Consider a physical system where a matrix describes multiple vibrational modes. How would diagonalization aid in understanding this system? Provide two distinct enhancements in understanding.

   Hint

   Think about how separating equations enhances clarity.