166 CHAPTER 7. CANONICAL FORMS

equals the number of Jordan blocks in the Jordan canonical form which are associatedwith λ . Recall the eigenspace is ker(λ I−A) .

11. 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.

12. Give an example of two nilpotent matrices which are not similar but have the sameminimum polynomial if possible.

13. Here is a matrix. Find its Jordan canonical form by directly finding the eigenvec-tors and generalized eigenvectors based on these to find a basis which will yield theJordan form. The eigenvalues are 1 and 2.

−3 −2 5 3−1 0 1 2−4 −3 6 4−1 −1 1 3

Why is it typically impossible to find the Jordan canonical form?

14. Let A be an n× n matrix and let J be its Jordan canonical form. Here F= R or C.Recall J is a block diagonal matrix having blocks Jk (λ ) down the diagonal. Each ofthese blocks is of the form

Jk (λ ) =

λ 1 0

λ. . .. . . 1

0 λ

Now for ε > 0 given, let the diagonal matrix Dε be given by

Dε =

1 0

ε

. . .

0 εk−1

Show that D−1

ε Jk (λ )Dε has the same form as Jk (λ ) but instead of ones down thesuper diagonal, there is ε down the super diagonal. That is Jk (λ ) is replaced with

λ ε 0

λ. . .. . . ε

0 λ

Now show that for A an n×n matrix, it is similar to one which is just like the Jordancanonical form except instead of the blocks having 1 down the super diagonal, it hasε .