7.7. EXERCISES 165
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 Ak and Bk both have the same rankfor all k ≥ 1. Does it follow that A,B are similar? What if it is not known that A,Bare nilpotent? Does it follow then?
4. (Review problem.) When we say a polynomial equals zero, we mean that all thecoefficients equal 0. If we assign a different meaning to it which says that a poly-nomial p(λ ) equals zero when it is the zero function, (p(λ ) = 0 for every λ ∈ F.)does this amount to the same thing? Is there any difference in the two definitions forordinary fields like Q? Hint: Consider for the field of scalars Z2, the integers mod 2and consider p(λ ) = λ
2 +λ .
5. Let A ∈L (V,V ) where V is a finite dimensional vector space with field of scalars F.Let p(λ ) be the minimum polynomial and suppose φ (λ ) is any nonzero polynomialsuch that φ (A) is not one to one and φ (λ ) has smallest possible degree such thatφ (A) is nonzero and not one to one. Show φ (λ ) must divide p(λ ).
6. Let A ∈L (V,V ) where V is a finite dimensional vector space with field of scalars F.Let p(λ ) be the minimum polynomial and suppose φ (λ ) is an irreducible polyno-mial with the property that φ (A)x = 0 for some specific x ̸= 0. Show that φ (λ ) mustdivide p(λ ) . Hint: First write p(λ ) = φ (λ )g(λ )+ r (λ ) where r (λ ) is either 0 orhas degree smaller than the degree of φ (λ ). If r (λ ) = 0 you are done. Suppose it isnot 0. Let η (λ ) be the monic polynomial of smallest degree with the property thatη (A)x = 0. Now use the Euclidean algorithm to divide φ (λ ) by η (λ ) . Contradictthe irreducibility of φ (λ ) .
7. Let A =
1 0 00 0 −10 1 0
Find the minimum polynomial for A.
8. 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 amonic polynomial 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 the φwk
(λ ) .Explain why this must be the minimum polynomial of A. Give a reasonably easyalgorithm for computing φv (λ ).
9. Here is a matrix. −7 −1 −1−21 −3 −370 10 10
Using the process of Problem 8 find the minimum polynomial of this matrix. Deter-mine whether it can be diagonalized from its minimum polynomial.
10. Let A be an n× n matrix with field of scalars C or more generally, the minimumpolynomial splits. Letting λ be an eigenvalue, show the dimension of the eigenspace