21.13.2. Relation to Characteristic Polynomial

Try these first

1. 1.

   Define the minimal polynomial.

   Hint

   Focus on what it means for a polynomial to be 'monic'.

2. 2.

   What is the characteristic polynomial's form?

   Hint

   Recall the definition of eigenvalues.

3. 3.

   What is the minimal polynomial of a matrix?

   • A polynomial of any form
   • A monic polynomial of least degree such that m(A) = 0
   • Only the characteristic polynomial

   Hint

   Remember that it must satisfy m(A) = 0.

4. 4.

   True or False: The minimal polynomial does not necessarily divide the characteristic polynomial.

   • True
   • False

   Hint

   Revisit the definitions of both polynomials.

5. 5.

   Given the matrix A = [[4, 1], [2, 3]], find its characteristic polynomial and minimal polynomial. Discuss their implications for the matrix's eigenstructure.

   Hint

   Use the determinant and polynomial properties to derive your results.

6. 6.

   Consider a control system that uses the minimal polynomial m(λ) = λ^2 + 5λ + 6. Analyze the roots and discuss system stability based on these roots.

   Hint

   Check if the roots are in the left half of the complex plane for stability.