Linear Algebra, Theory And Applications, 2012a

262 LINEAR TRANSFORMATIONS CANONICAL FORMS
10.6 Exercises
1. In the discussion of Nilpotent transformations, it was asserted that if two n×n matrices
A, B are similar, then A k is also similar to B k . Why is this so? If two matrices are
similar, why must they have the same rank?
2. If A, B are both invertible, then they are both row equivalent to the identity matrix.
Are they necessarily similar? Explain.
3. Suppose you have two nilpotent matrices A, B and A k and B k both have the same
rank for all k ≥ 1. Does it follow that A, B are similar? What if it is not known that
A, B are nilpotent? Does it follow then?
4. When we say a polynomial equals zero, we mean that all the coefficients equal 0. If
we assign a different meaning to it which says that a polynomial
p (λ) =
n∑
a k λ k =0,
k=0
when the value of the polynomial equals zero whenever a particular value of λ ∈ F
is placed in the formula for p (λ) , can the same conclusion be drawn? Is there any
difference in the two definitions for ordinary fields like Q? Hint: Consider Z 2 , the
integers mod 2.
5. Let A ∈L(V,V )whereV is a finite dimensional vector space with field of scalars F.
Let p (λ) be the minimal polynomial and suppose φ (λ) is any nonzero polynomial such
that φ (A) is not one to one and φ (λ) has smallest possible degree such that φ (A) is
nonzero and not one to one. Show φ (λ) must divide p (λ).
6. Let A ∈L(V,V )whereV is a finite dimensional vector space with field of scalars F.
Let p (λ) be the minimal polynomial and suppose φ (λ) is an irreducible polynomial
with the property that φ (A) x = 0 for some specific x ≠0. Show that φ (λ) must
divide p (λ) . Hint: First write
p (λ) =φ (λ) g (λ)+r (λ)
where r (λ) is either 0 or has degree smaller than the degree of φ (λ). If r (λ) =0you
are done. Suppose it is not 0. Let η (λ) be the monic polynomial of smallest degree
with the property that η (A) x =0. Now use the Euclidean algorithm to divide φ (λ)
by η (λ) . Contradict the irreducibility of φ (λ) .
7. Suppose A is a linear transformation and let the characteristic polynomial be
det (λI − A) =
q∏
φ j (λ) n j
where the φ j (λ) are irreducible. Explain using Corollary 8.3.11 why the irreducible
factors of the minimal polynomial are φ j (λ) and why the minimal polynomial is of
the form
q∏
φ j (λ) r j
where r j ≤ n j . You can use the Cayley Hamilton theorem if you like.
j=1
j=1