9.6. EXERCISES 253

19. Let A be an n× n matrix and let J be its Jordan canonical form. Recall J is a blockdiagonal matrix having blocks Jk (λ) down the diagonal. Each of these blocks is ofthe 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.

20. Let A be in L (V, V ) and suppose that Apx ̸= 0 for some x ̸= 0. Show that Apek ̸= 0for some ek ∈ {e1, · · · , en} , a basis for V . If you have a matrix which is nilpotent,(Am = 0 for some m) will it always be possible to find its Jordan form? Describe howto do it if this is the case. Hint: First explain why all the eigenvalues are 0. Thenconsider the way the Jordan form for nilpotent transformations was constructed in theabove.

21. Suppose A is an n×n matrix and that it has n distinct eigenvalues. How do the mini-mal polynomial and characteristic polynomials compare? Determine other conditionsbased on the Jordan Canonical form which will cause the minimal and characteristicpolynomials to be different.

22. Suppose A is a 3× 3 matrix and it has at least two distinct eigenvalues. Is it possiblethat the minimal polynomial is different than the characteristic polynomial?

23. If A is an n×n matrix of entries from a field of scalars and if the minimal polynomialof A splits over this field of scalars, does it follow that the characteristic polynomialof A also splits? Explain why or why not.

24. Show that if two n × n matrices A,B are similar, then they have the same minimalpolynomial and also that if this minimal polynomial is of the form p (λ) =

∏si=1 ϕi (λ)

ri

where the ϕi (λ) are irreducible and monic, then ker (ϕi (A)ri) and ker (ϕi (B)

ri) havethe same dimension. Why is this so? This was what was responsible for the blockscorresponding to an eigenvalue being of the same size.