SWILA PROBLEM SET #7 SECTION 5.6 MONDAY, JUNE 27

Goals for Problem Set: • Learn what the invariant factors of a linear operator are. • Figure out how invariant factors are connected to the minimal polynomial and the characteristic polynomial. • Determine how cyclic subspaces and companion matrices are connected to the rational canonical form. Instructions: The below problems are split up by difficulty level: Easiest, Mediumest, and Hardest. Within each difficulty level, the problems are ordered by how good I think they are to do. In other words, the problems that I think are best to do are listed earlier within each difficulty level. I recommend that you try problems from various sections or from sections that are new to you. Warnings: Labeling the difficulty of problems was performed rather imprecisely. Thus, some problems will be mislabeled. The same holds with ordering problems by how good they are to do. I also didn’t really proofread these problems so there may be typos, and I don’t know how to completely solve all of them.

I like the first 4 “Easiest” problems. I like the first 3 “Mediumest” problems. I think the first 3 “Hardest” problems look interesting. Easiest Problems Exercise 1.0.1. (5.6)(Petersen, pg. 148) Consider the following matrices in rational canonical form:     0 0 0 0 0 0 0 0   1 0 0 0    , A2 =  1 0 0 0  , A1 =   0 1 0 0   0 1 0 0  0 0 1 0 0 0 0 0     0 0 0 0 0 0 0 0  1 0 0 0   1 0 0 0     A3 =   0 0 0 0  , A4 =  0 0 0 0  . 0 0 1 0 0 0 0 0 Note each matrix’s characteristic polynomial is t4 . Find the invariant factors of each matrix. Conclude that the minimal polynomial and the characteristic polynomial does not uniquely determine a matrix up to similarity. Hint: Each matrix is already in rational canonical form. 1

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Exercise 1.0.2. (5.6)(Petersen, pg. 149, #2) Let A ∈ Mn (R), and n ≥ 2. (a) Show that when n is odd, then it is not possible to have p1 (t) = t2 + 1. (b) Show by example that one can have p1 (t) = t2 + 1 for all even n. (c) Show by example that one can have p1 (t) = t3 + t for all odd n. Exercise 1.0.3. (5.6) Determine whether each of the following matrices is in rational canonical form. If so, identify the invariant factors, the minimal polynomial, and the characteristic polynomial. (a)   0 0 4 0 0  1 0 0 0 0     0 1 −3 0 0  .    0 0 0 0 −4  0 0 0 1 4 (b)   0 0 4 0 0  1 0 0 0 0     0 1 −3 0 0  .    0 0 0 0 −4  0 0 0 1 −4 Note that t3 + 3t2 − 4 = (t + 2)2 (t − 1). Exercise 1.0.4. (5.6)(Petersen, pg. 149, #3) If T : V → V is an operator on a 2-dimensional space, then either p1 = µT = χT or T = λ1V for some λ ∈ F. Exercise 1.0.5. (5.6)(Petersen, pg. 149, #1) What are the similarity invariants for a companion matrix Ap ? Exercise 1.0.6. (5.6) This exercise will fill in details of the proof of the rational canonical form. Fix a vector space V over a field F. (a) Fix x0 ∈ V , a linear operator T : V → V , and p(t) ∈ F[t]. Let Cx0 = {f (T )x0 : f (t) ∈ F[t]} be the cyclic subspace generated by x0 . If f (T )x0 = 0, prove that f (T )y = 0 for all y ∈ Cx0 . Hint: Recall f (T ) ◦ g(T ) = g(T ) ◦ f (T ) for all f, g ∈ F[t]. (b) Suppose p(t) ∈ F[t] is a polynomial of degree d and T : V → V a linear operator. If there exists x ∈ V such that {x, T x, . . . , T d−1 x} is linearly independent and p(T ) = 0, prove that p(t) = mT (t), i.e., p(t) is the minimal polynomial of T . Hint: Recall that mT (t) is defined to be the polynomial of lowest degree so that mT (T ) = 0. (c) Justify that l ≤ m in the middle of the proof of the rational canonical form.

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Mediumest Problems Exercise 1.0.7. (5.6)(Petersen, pg. 148) (Rational canonical form of projections) Suppose dim(V ) = n, and let T : V → V be a projection, i.e., T 2 = T . (a) Suppose that T 6= 0V and T 6= 1V . Prove that the minimal of T is t2 − t.  polynomial  0 0 (b) Note the companion matrix of the polynomial t2 − t is . Find the maximum 1 1 number of blocks (in terms of n) of the form At2 −t in the rational canonical form of T . (c) Justify that each of the blocks not equal to At2 −t in the rational canonical form of T has to be (1) or each of the blocks not equal to At2 −t has to be (0). As a consequence, identify the two types of rational canonical forms for projections. Exercise 1.0.8. (5.6)(Petersen, pg. 149, #4) If T : V → V is an operator on a 3-dimensional space, then exactly one of the following holds: • p1 = µT = χT . • p1 = (t − α)(t − β) and p2 = (t − β), some α, β ∈ F. • T = λ1V , some λ ∈ F. Exercise 1.0.9. (5.6)(Petersen, pg. 149, #5) Let T : V → V be a linear operator on a finite-dimensional space. Show that V = Cx for some x ∈ V if and only if µT = χT . Exercise 1.0.10. (5.6) (Petersen, pg. 149, #13) Recall the Cayley-Hamilton Theorem states that χT (T ) = 0 for all linear operators T : V → V . Prove this theorem using the rational canonical form. Exercise 1.0.11. (Hoffman, pg. 242, #7)(5.6) Find the minimal polynomials and the rational canonical forms of each of the following matrices       c 0 −1 0 −1 −1 cos θ sin θ  1 0 0 ,  0 c 1 , . − sin θ cos θ −1 1 c −1 0 0 Exercise 1.0.12. (5.6) This exercise will fill in details of the proof of the rational canonical form. Fix a vector space V over a field F. (a) In the proof of uniqueness of the rational canonical form, justify why it suffices to show the minimal polynomials of A22 and A022 coincide. (b) Fix A ∈ Mn (F) and S ∈ GLn (F). Prove that for all p(t) ∈ F[t], p(SAS −1 ) = Sp(A)S −1 . Exercise 1.0.13. (5.6) Prove the following proposition from the notes: For matrices A, B ∈ Mn (F), the following are equivalent: (a) A is similar to B. (b) A and B have the same invariant factors. (c) A and B have the same rational canonical form. Exercise 1.0.14. (5.6)(Petersen, pg. 149, #12) Fix λ ∈ F. Assume that T : V → V satisfies (T − λ1V )k = 0, for some k > 1, but (T − λ1V )k−1 6= 0. Show that ker(T − λ1V ) is neither {0} nor V . Show that ker(T − λ1V ) does not have a complement in V that is T invariant.

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Exercise 1.0.15. (Hoffman, pg. 242, #9)(5.6) Let A be the real matrix   1 3 3 1 3 . A= 3 −3 −3 −5 Find an invertible matrix P such that P −1 AP is in rational canonical form. Hardest Problems Exercise 1.0.16. (5.6)(Petersen, pg. 149, #6) Consider two companion matrices Ap and Aq . Show that the invariant factors for the block diagonal matrix   Ap 0 0 Aq are p1 = lcm{p, q} and p2 = gcd{p, q}. Exercise 1.0.17. (Hoffman, pg. 243, #13)(5.6) Let A be an n × n matrix with complex entries. Prove that if every eigenvalue of A is real, then A is similar to a matrix with real entries. Exercise 1.0.18. (5.6)(Petersen, pg. 149, #11) Let T : V → V be a linear operator on a finite-dimensional vector space. Use the rational canonical form to show that tr(T ) = −an−1 , where χT (t) = tn + an−1 tn−1 + · · · + a0 . Exercise 1.0.19. (Hoffman, pg. 243, #12)(5.6) Let F be a subfield of C, and let A and B be n × n matrices over F. Prove that if A and B are similar over C, then they are similar over F. Hint: Prove that the rational canonical form of A is the same whether A is viewed as a matrix over F or a matrix over C; likewise for B. Exercise 1.0.20. (Hoffman, pg. 243, #11)(5.6) Let F be a subfield of C. (It may be possible that F can be taken to be any field.) Prove that A and B are 3 × 3 matrices over F, a necessary and sufficient condition that A and B be similar over F is that they have the same characteristic polynomial and the same minimal polynomial. Give an example which shows that this is false for 4 × 4 matrices. Exercise 1.0.21. (5.6)(Petersen, pg. 149, #8) Show that A, B ∈ Mn (F) are similar if and only if rank(p(A)) = rank(p(B)) for all p ∈ F[t]. (Recall that matrices A, B are equivalent if and only if A = P BQ−1 for some P, Q ∈ GLn (F). Recall that two matrices have the same rank if and only if they are equivalent and that equivalent matrices certainly need not be similar. This is what makes the exercise interesting.) Exercise 1.0.22. (5.6)(Petersen, pg. 149, #9) The previous condition can be made into a checkable condition: Show that A, B ∈ Mn (F) are similar if and only if χA = χB and rank(p(A)) = rank(p(B)) for all p that divide χA . (Using that as χA has a unique prime factorization this means that we only have to check a finite number of conditions.) Exercise 1.0.23. (Hoffman, pg. 243, #14)(5.6) Let T be a linear operator on a finitedimensional vector space V . Prove that there exists a vector x ∈ V with the following property: If f is a polynomial and f (T )x = 0, then f (T ) = 0. (Such a vector x is called a separating vector for the algebra of polynomials in T .) When T has a cyclic vector, give a direct proof that any cyclic vector is a separating vector for the algebra of polynomials in T .

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Exercise 1.0.24. (Hoffman, pg. 243, #16)(5.6) Let A be an n × n matrix with real entries 2 such that = 0. Prove that n is even, and A is similar over R to a matrix of the block  A +I 0 −Ik form . Ik 0 Exercise 1.0.25. (5.6)(Petersen, pg. 149, #7) Is it possible to find the similarity invariants for   Cp 0 0  0 C q 0 ? 0 0 Cr Note that you can easily find p1 = lcm{p, q, r}, so the issue is whether it is possible to decide what p2 should be. Exercise 1.0.26. (5.6)(Petersen, pg. 149, #10) Show that any linear map with the property that χT (t) = (t − λ1 ) · · · (t − λn ) ∈ F[t] for λ1 , . . . , λn ∈ F has an upper triangular matrix representation. Exercise 1.0.27. (5.6) This exercise will fill in details of the proof of the rational canonical form. Fix a vector space V over a field F. (a) In the proof of the rational canonical form, we found a T -invariant complementary subspace M of Cx1 . We chose l to be the largest dimension of a cyclic subspace of M , say Cx2 , and we chose p2 to be a polynomial of degree l such that p2 (T )(x2 ) = 0. We then wrote p1 = q · p2 + r, where deg(r) < deg(p2 ). Prove that r = 0, and hence p2 divides p1 . (b) In the proof of uniqueness of the rational canonical form, prove that     p(A11 ) 0 p(A11 ) 0 −1 S. =S 0 p(A022 ) 0 p(A22 ) References [1] Kenneth Hoffman and Ray Kunze. Linear algebra. 2nd ed. Prentice Hall, Englewood Cliffs, NJ, 1971. [2] Peter Petersen. Linear algebra. Los Angeles, CA, 2000. http://www.calpoly.edu/~ jborzell/Courses/Year%2010-11/Fall%202010/ Petersen-Linear Algebra-Math 306.pdf.