21.13.3. Application

Try these first

1. 1.

   What is the definition of a minimal polynomial?

   Hint

   Think about the polynomial's role concerning the matrix.

2. 2.

   True or False: The minimal polynomial must always be of higher degree than the characteristic polynomial.

   Hint

   Consider their relationship.

3. 3.

   What defines a minimal polynomial?

   Hint

   Think about the polynomial's role in matrix equations.

4. 4.

   True or False: The minimal polynomial only applies to square matrices.

   • True
   • False

   Hint

   Recall the definition of a minimal polynomial.

5. 5.

   You encounter a 3x3 matrix with a characteristic polynomial of (λ-1)(λ-2)². Determine the minimal polynomial and discuss its implications on stability.

   Hint

   Reflect on the meaning of repeated roots in the context of Jordan forms.

6. 6.

   Consider a matrix with eigenvalues 3, 4, and a repeated eigenvalue of 5. How would this affect its minimal polynomial and theorize on the potential behavior of the system modeled by this matrix?

   Hint

   Think about how multiple eigenvalues affect system dynamics, particularly in control systems.