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.)

Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter5: Orthogonality
Section5.4: Orthogonal Diagonalization Of Symmetric Matrices
Problem 27EQ
icon
Related questions
Question

Linear algebra: please solve q18 correctly and handwritten. Theoram is also attached

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).
Transcribed Image Text: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.)
Transcribed Image Text: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.)
Expert Solution
steps

Step by step

Solved in 4 steps with 4 images

Blurred answer
Recommended textbooks for you
Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning
Calculus For The Life Sciences
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,