21.13.1. Definition

Try these first

1. 1.

   What is the minimal polynomial of a matrix?

   Hint

   Think about polynomials and matrices.

2. 2.

   Does the minimal polynomial divide the characteristic polynomial?

   Hint

   Consider how these polynomials relate to eigenvalues.

3. 3.

   What is the degree of the minimal polynomial related to?

   • It is always greater than the degree of the characteristic polynomial
   • It cannot be less than the degree of the characteristic polynomial
   • It can be equal to or less than the degree of the characteristic polynomial

   Hint

   Think about how polynomials relate in linear algebra.

4. 4.

   Is it true that the minimal polynomial must be a factor of the characteristic polynomial?

   • True
   • False

   Hint

   Recall their mathematical relationship.

5. 5.

   Given the matrix A = [[1, 2],[0, 1]], find its minimal polynomial and explain your reasoning.

   Hint

   Calculate the eigenvalues and their multiplicities.

6. 6.

   If a matrix has a minimal polynomial of m(x) = (x - 3)(x - 2)^2, what can you infer about its eigenvalues and their respective multiplicities?

   Hint

   Review the factors and their powers.