9.6. EXERCISES 251
Find the minimal polynomial for A.
9. Suppose A is an n × n matrix and let v be a vector. Consider the A cyclic set ofvectors
{v, Av, · · · , Am−1v
}where this is an independent set of vectors but Amv is
a linear combination of the preceding vectors in the list. Show how to obtain a monicpolynomial of smallest degree, m, ϕv (λ) such that
ϕv (A)v = 0
Now let {w1, · · · ,wn} be a basis and let ϕ (λ) be the least common multiple of theϕwk
(λ) . Explain why this must be the minimal polynomial of A. Give a reasonablyeasy algorithm for computing ϕv (λ).
10. Here is a matrix. −7 −1 −1
−21 −3 −3
70 10 10
Using the process of Problem 9 find the minimal polynomial of this matrix. It turnsout the characteristic polynomial is λ3.
11. Find the minimal polynomial for
A =
1 2 3
2 1 4
−3 2 1
by the above technique. Is what you found also the characteristic polynomial?
12. Let A be an n × n matrix with field of scalars C. Letting λ be an eigenvalue, showthe dimension of the eigenspace equals the number of Jordan blocks in the Jordancanonical form which are associated with λ. Recall the eigenspace is ker (λI −A) .
13. For any n × n matrix, why is the dimension of the eigenspace always less than orequal to the algebraic multiplicity of the eigenvalue as a root of the characteristicequation? Hint: Note the algebraic multiplicity is the size of the appropriate blockin the Jordan form.
14. Give an example of two nilpotent matrices which are not similar but have the sameminimal polynomial if possible.
15. Use the existence of the Jordan canonical form for a linear transformation whoseminimal polynomial factors completely to give a proof of the Cayley Hamilton theoremwhich is valid for any field of scalars. Hint: First assume the minimal polynomialfactors completely into linear factors. If this does not happen, consider a splitting fieldof the minimal polynomial. Then consider the minimal polynomial with respect tothis larger field. How will the two minimal polynomials be related? Show the minimalpolynomial always divides the characteristic polynomial.
16. Here is a matrix. Find its Jordan canonical form by directly finding the eigenvectorsand generalized eigenvectors based on these to find a basis which will yield the Jordanform. The eigenvalues are 1 and 2.
−3 −2 5 3
−1 0 1 2
−4 −3 6 4
−1 −1 1 3