21.13. Minimal Polynomial

Try these first

1. 1.

   Define minimal polynomial in your own words.

   Hint

   Think about what it means for a polynomial to 'annihilate' a matrix.

2. 2.

   What is a monic polynomial?

   Hint

   What do you recall about polynomial coefficients?

3. 3.

   What is the minimal polynomial of a matrix?

   • A polynomial that always has distinct roots.
   • The polynomial of least degree that annihilates the matrix.
   • A polynomial that has a non-zero leading coefficient.

   Hint

   Focus on the purpose of the polynomial.

4. 4.

   True or False: The minimal polynomial of a matrix can never be of higher degree than the characteristic polynomial.

   • True
   • False

   Hint

   Think about the degree relationship between the two polynomials.

5. 5.

   Find the minimal polynomial for the matrix A = [[3, 1, 0], [0, 3, 1], [0, 0, 3]]. What does this tell you about A?

   Hint

   Calculate the characteristic polynomial first.

6. 6.

   Evaluate the implications of a minimal polynomial m(x) = x^3 for a 3x3 matrix. What structure does this imply for the eigenvalues?

   Hint

   Consider the multiplicity of the eigenvalues.