7.7. EXERCISES 167
15. 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 inthe above.
16. Show that if two n× n matrices A,B are similar, then they have the same mini-mum polynomial and also that if this minimum polynomial is of the form p(λ ) =∏
si=1 φ i (λ )
ri where the φ i (λ ) are irreducible and monic, then ker(φ i (A)ri) and
ker(φ i (B)ri) have the same dimension. Why is this so? This was what was re-
sponsible for the blocks corresponding to an eigenvalue being of the same size.
17. Show that each cyclic set β x is associated with a monic polynomial ηx (λ ) suchthat ηx (A)(x) = 0 and this polynomial has smallest possible degree such that thishappens. Show that the cyclic sets β xi
can be arranged such that ηxi+1(λ )/ηxi
(λ ).
18. Show that if A is a complex n×n matrix, then A and AT are similar. Hint: Considera Jordan block. Note that 0 0 1
0 1 01 0 0
λ 1 0
0 λ 10 0 λ
0 0 1
0 1 01 0 0
=
λ 0 01 λ 00 1 λ
19. (Extra important) Let A be an n× n matrix. The trace of A, trace(A) is defined as
∑i Aii. It is just the sum of the entries on the main diagonal. Show trace(A) =trace
(AT). Suppose A is m×n and B is n×m. Show that trace(AB) = trace(BA) .
Now show that if A and B are similar n×n matrices, then trace(A) = trace(B). Recallthat A is similar to B means A = S−1BS for some matrix S.
20. (Extra important) If A is an n×n matrix and the minimum polynomial splits in F thefield of scalars, show that trace(A) equals the sum of the eigenvalues listed accordingto multiplicity according to number of times they occur in the Jordan form.
21. Let A be a linear transformation defined on a finite dimensional vector space V . Letthe minimum polynomial be ∏
qi=1 φ i (λ )
mi and let(
βivi
1, · · · ,β i
viri
)be the cyclic sets
such that{
βivi
1, · · · ,β i
viri
}is a basis for ker(φ i (A)
mi). Let v = ∑i ∑ j vij. Now let q(λ )
be any polynomial and suppose that q(A)v = 0. Show that it follows q(A) = 0.Hint: First consider the special case where a basis for V is
{x,Ax, · · · ,An−1x
}and
q(A)x = 0.
22. Find the minimum polynomial for A=
1 2 32 1 4−3 2 1
assuming the field of scalars
is the rational numbers.
23. Show, using the rational root theorem, the minimum polynomial for A in the aboveproblem is irreducible with respect to Q. Letting the field of scalars be Q find therational canonical form and a similarity transformation which will produce it.