306 3. ³ [34] $[6] 5. 1 -1 3 7. 9. 11. -6-1 2 320 -14 -2 5 3-1-1 -12 0 5 4 -2 -1 24 4 1-1 3 4. 13-16 9-11 6. 8. 2 2 33 12. -2-2 10. -7 4-3 83 3 32-16 13 6441 461 4164 14 46 4 Chapter 4 The Eigenvalue Problem Exercises 20-23 illustrate the Cayley-Hamilton theo- rem, which states that if p(t) is the characteristic poly- nomial for A, then p(A) is the zero matrix. (As in Ex- ercise 18, p (A) is the (n × n) matrix that comes from substituting A fort in p(t).) In Exercises 20-23, verify that p(A) = O for the given matrix A. 20. A in Exercise 3 21. A in Exercise 4 23. A in Exercise 13 22. A in Exercise 9 with Ax = λx, x = 0.] 16. Prove property (c) of Theorem 11. 17. Complete the proof of property (a) of Theorem 11. 18. Let q(t) = t³ - 21² - t + 2; and for any (nx n) matrix H, define the matrix polynomial q (H) by q(H) = H³ - 2H² — H+21, where I is the (n × n) identity matrix. a) Prove that if λ is an eigenvalue of H, then the number q (2) is an eigenvalue of the matrix q(H). [Hint: Suppose that Hx = λx, where x = 0, and use Theorem 11 to evaluate q (H)x.] b) Use part a) to calculate the eigenvalues of q (A) and q (B), where A and B are from Exercises 7 and 8, respectively. 19. With q (t) as in Exercise 18, verify that q(C) is the zero matrix, where C is from Exercise 9. (Note that q(t) is the characteristic polynomial for C. See Ex- ercises 20-23.) A = -an-1-an-2 ... a₁-ao 0 0 1 0 0 0 1 0 0 0 1 0 a) For n = 2 and for n = 3, show that det(AtI) = (-1)" q(t). THEOREM 11 Let A be an (nx n) matrix, and let λ be an eigenvalue of A. Then: (a) is an eigenvalue of Ak, k = 2, 3, .... (b) If A is nonsingular, then 1/2 is an eigenvalue of A-¹. (c) Ifa is any scalar, then λ + a is an eigenvalue of A + aI. Proof Property (a) is proved by induction, and we begin with the case k = 2. Suppose that > is an eigenvalue of A with an associated eigenvector, x. That is, Ax = λx, x ± 0. Multiplying both sides of Eq. (2) by the matrix A gives A(Ax) = A(2x) A²x = λ(Ax) A²x = λ(2x) A²x = 1²x, x‡0. (2) Thus ² is an eigenvalue of A² with a corresponding eigenvector, x. In the exercises the reader is asked to finish proof of property (a) and prove prop- erties (b) and (c) of Theorem 11. (Note: As the proof of Theorem 11 will demonstrate, if x is any eigenvector of A, then x is also an eigenvector of Ak, A-1, and A + αI.)