Exercises 5B 1. 1  Suppose 𝑇 ∈ L(𝑉). Prove that 9 is an eigenvalue of 𝑇2 if and only if 3 or −3 is an eigenvalue of 𝑇. 2. 2  Suppose 𝑉 is a complex vector space and 𝑇 ∈ L(𝑉) has no eigenvalues. Prove that every subspace of 𝑉 invariant under 𝑇 is either {0} or infinite- dimensional. 3. 3  Suppose 𝑛 is a positive integer and 𝑇 ∈ L(𝐅𝑛) is defined by 𝑇(𝑥1,...,𝑥𝑛)=(𝑥1 +⋯+𝑥𝑛,...,𝑥1 +⋯+𝑥𝑛). (a) Findalleigenvaluesandeigenvectorsof𝑇. (b) Findtheminimalpolynomialof𝑇. The matrix of 𝑇 with respect to the standard basis of 𝐅𝑛 consists of all 1’s. 4. 4  Suppose𝐅 = 𝐂,𝑇 ∈ L(𝑉),𝑝 ∈ 𝒫(𝐂),and𝛼 ∈ 𝐂. Provethat𝛼isan eigenvalue of 𝑝(𝑇) if and only if 𝛼 = 𝑝(𝜆) for some eigenvalue 𝜆 of 𝑇. 5. 5  Give an example of an operator on 𝐑2 that shows the result in Exercise 4 does not hold if 𝐂 is replaced with 𝐑. 6. 6  Suppose 𝑇 ∈ L(𝐅2) is defined by 𝑇(𝑤, 𝑧) = (−𝑧, 𝑤). Find the minimal polynomial of 𝑇. 7. 7  (a) Give an example of 𝑆, 𝑇 ∈ L(𝐅2) such that the minimal polynomial of 𝑆𝑇 does not equal the minimal polynomial of 𝑇𝑆. (b) Suppose 𝑉 is finite-dimensional and 𝑆, 𝑇 ∈ L(𝑉). Prove that if at least one of 𝑆, 𝑇 is invertible, then the minimal polynomial of 𝑆𝑇 equals the minimal polynomial of 𝑇𝑆. Hint: Show that if 𝑆 is invertible and 𝑝 ∈ 𝒫(𝐅), then 𝑝(𝑇𝑆) = 𝑆−1𝑝(𝑆𝑇)𝑆. 8. 8  Suppose 𝑇 ∈ L(𝐑2) is the operator of counterclockwise rotation by 1∘. Find the minimal polynomial of 𝑇. Because dim 𝐑2 = 2, the degree of the minimal polynomial of 𝑇 is at most 2. Thus the minimal polynomial of 𝑇 is not the tempting polynomial 𝑥180 + 1, even though 𝑇180 = −𝐼. 9. 9  Suppose 𝑇 ∈ L(𝑉) is such that with respect to some basis of 𝑉, all entries of the matrix of 𝑇 are rational numbers. Explain why all coefficients of the minimal polynomial of 𝑇 are rational numbers. 10. 10  Suppose 𝑉 is finite-dimensional, 𝑇 ∈ L(𝑉), and 𝑣 ∈ 𝑉. Prove that span(𝑣,𝑇𝑣,...,𝑇𝑚𝑣) = span(𝑣,𝑇𝑣,...,𝑇dim𝑉−1𝑣) for all integers 𝑚 ≥ dim𝑉 − 1. 11. 11  Suppose 𝑉 is a two-dimensional vector space, 𝑇 ∈ L(𝑉), and the matrix of 𝑇 with respect to some basis of 𝑉 is ( 𝑎 𝑐 ). 𝑏𝑑 (a) Show that 𝑇2−(𝑎+𝑑)𝑇+(𝑎𝑑−𝑏𝑐)𝐼=0. (b) Show that the minimal polynomial of 𝑇 equals ⎧{𝑧 − 𝑎 if 𝑏 = 𝑐 = 0 and 𝑎 = 𝑑, ⎨{⎩𝑧2 − (𝑎 + 𝑑)𝑧 + (𝑎𝑑 − 𝑏𝑐) otherwise. 12. 12  Define 𝑇 ∈ L(𝐅𝑛) by 𝑇(𝑥1, 𝑥2, 𝑥3, ..., 𝑥𝑛) = (𝑥1, 2𝑥2, 3𝑥3, ..., 𝑛𝑥𝑛). Find the minimal polynomial of 𝑇. 13. 13  Suppose 𝑇 ∈ L(𝑉) and 𝑝 ∈ 𝒫(𝐅). Prove that there exists a unique 𝑟 ∈ 𝒫(𝐅) such that 𝑝(𝑇) = 𝑟(𝑇) and deg𝑟 is less than the degree of the minimal polynomial of 𝑇. 14. 14  Suppose 𝑉 is finite-dimensional and 𝑇 ∈ L(𝑉) has minimal polynomial 4 + 5𝑧 − 6𝑧2 − 7𝑧3 + 2𝑧4 + 𝑧5. Find the minimal polynomial of 𝑇−1. 15. 15  Suppose 𝑉 is a finite-dimensional complex vector space with dim 𝑉 > 0 and 𝑇 ∈ L(𝑉). Define 𝑓∶ 𝐂 → 𝐑 by 𝑓 (𝜆) = dim range(𝑇 − 𝜆𝐼). Prove that 𝑓 is not a continuous function. 16. 16  Suppose 𝑎0, ..., 𝑎𝑛 − 1 ∈ 𝐅. Let 𝑇 be the operator on 𝐅𝑛 whose matrix (with respect to the standard basis) is ⎛⎜0 −𝑎0 ⎞⎟ ⎜10 −𝑎1 ⎟ ⎜1⋱−𝑎2 ⎟. ⎜ ⋱ ⋮ ⎟ ⎜ 0 −𝑎𝑛−2 ⎟ ⎝ 1 −𝑎𝑛−1 ⎠ Here all entries of the matrix are 0 except for all 1’s on the line under the diagonal and the entries in the last column (some of which might also be 0). Show that the minimal polynomial of 𝑇 is the polynomial 𝑎0 +𝑎1𝑧+⋯+𝑎𝑛−1𝑧𝑛−1 +𝑧𝑛. The matrix above is called the companion matrix of the polynomial above. This exercise shows that every monic polynomial is the minimal polynomial of some operator. Hence a formula or an algorithm that could produce exact eigenvalues for each operator on each 𝐅𝑛 could then produce exact zeros for each polynomial [by 5.27(a)]. Thus there is no such formula or algorithm. However, efficient numeric methods exist for obtaining very good approximations for the eigenvalues of an operator. 17. 17  Suppose 𝑉 is finite-dimensional, 𝑇 ∈ L(𝑉), and 𝑝 is the minimal polynomial of 𝑇. Suppose 𝜆 ∈ 𝐅. Show that the minimal polynomial of 𝑇 − 𝜆𝐼 is the polynomial 𝑞 defined by 𝑞(𝑧) = 𝑝(𝑧 + 𝜆). 18. 18  Suppose 𝑉 is finite-dimensional, 𝑇 ∈ L(𝑉), and 𝑝 is the minimal polynomial of 𝑇. Suppose 𝜆 ∈ 𝐅\{0}. Show that the minimal polynomial of 𝜆𝑇 is the polynomial 𝑞 defined by 𝑞(𝑧) = 𝜆deg 𝑝 𝑝( 𝜆𝑧 ). Linear Algebra Done Right, fourth edition, by Sheldon Axler 19. 19 Suppose 𝑉 is finite-dimensional and 𝑇 ∈ L(𝑉). Let E be the subspace of L(𝑉) defined by Prove that dim E equals the degree of the minimal polynomial of 𝑇. E = {𝑞(𝑇) ∶ 𝑞 ∈ 𝒫(𝐅)}. 20. 20 Suppose 𝑇 ∈ L(𝐅4) is such that the eigenvalues of 𝑇 are 3, 5, 8. Prove that (𝑇−3𝐼)2(𝑇−5𝐼)2(𝑇−8𝐼)2 =0. 21. 21  Suppose 𝑉 is finite-dimensional and 𝑇 ∈ L(𝑉). Prove that the minimal polynomial of 𝑇 has degree at most 1 + dim range 𝑇. If dim range 𝑇 < dim 𝑉 − 1, then this exercise gives a better upper bound than 5.22 for the degree of the minimal polynomial of 𝑇. 22. 22  Suppose 𝑉 is finite-dimensional and 𝑇 ∈ L(𝑉). Prove that 𝑇 is invertible if and only if 𝐼 ∈ span(𝑇,𝑇2,...,𝑇dim𝑉). 23. 23  Suppose 𝑉 is finite-dimensional and 𝑇 ∈ L(𝑉). Let 𝑛 = dim 𝑉. Prove that if 𝑣 ∈ 𝑉, then span(𝑣,𝑇𝑣,...,𝑇𝑛−1𝑣) is invariant under 𝑇. 24. 24  Suppose 𝑉 is a finite-dimensional complex vector space. Suppose 𝑇 ∈ L(𝑉) is such that 5 and 6 are eigenvalues of 𝑇 and that 𝑇 has no other eigenvalues. Prove that (𝑇−5𝐼) dim 𝑉−1 (𝑇−6𝐼) dim 𝑉−1 =0. 25. 25  Suppose 𝑉 is finite-dimensional, 𝑇 ∈ L(𝑉), and 𝑈 is a subspace of 𝑉 that is invariant under 𝑇. (a) Prove that the minimal polynomial of𝑇 is a polynomial multiple of the minimal polynomial of the quotient operator 𝑇/𝑈. (b) Prove that (minimal polynomial of 𝑇|𝑈) × (minimal polynomial of 𝑇/𝑈) is a polynomial multiple of the minimal polynomial of 𝑇. The quotient operator 𝑇/𝑈 was defined in Exercise 38 in Section 5A. 26. 26  Suppose 𝑉 is finite-dimensional, 𝑇 ∈ L(𝑉), and 𝑈 is a subspace of 𝑉 that is invariant under 𝑇. Prove that the set of eigenvalues of 𝑇 equals the union of the set of eigenvalues of 𝑇|𝑈 and the set of eigenvalues of 𝑇/𝑈. 27. 27  Suppose 𝐅 = 𝐑, 𝑉 is finite-dimensional, and 𝑇 ∈ L(𝑉). Prove that the minimal polynomial of 𝑇𝐂 equals the minimal polynomial of 𝑇. The complexification 𝑇𝐂 was defined in Exercise 33 of Section 3B. 28. 28  Suppose 𝑉 is finite-dimensional and 𝑇 ∈ L(𝑉). Prove that the minimal polynomial of 𝑇′ ∈ L(𝑉′) equals the minimal polynomial of 𝑇. The dual map 𝑇′ was defined in Section 3F. 29. 29  Show that every operator on a finite-dimensional vector space of dimension at least two has an invariant subspace of dimension two. Exercise 6 in Section 5C will give an improvement of this result when 𝐅 = 𝐂.