In Exercises 1-18, s = 1+2i, u = 3-2i, v = 4+i, w = 2-i, and z = 1+i. In each exercise, perform the indicated calculation and express the result in the form a + ib. 1. ū 4. Z + w 7. vv 10. z²w 13. u/v 16. (w + v)/u Find the eigenvalues and the eigenvectors for the matri- ces in Exercises 19-24. (For the matrix in Exercise 24, one eigenvalue is λ = 1 + 5i.) 19. 68 -1 2 27. x = 2. Z 5. u + ū 8. uv 11. ūw² 14. v/u² 17. w + iz 29. x = In Exercises 27-30, calculate ||x||. [ ] 1+i 2 1 3. u + v 6. s-s 9. s2 – w 12. s(u² + v) 15. s/z 18. s - iw 2i 3+i 20. 2⁰ [44] 28. x = 30. x = 3+i 2-i 2i 1-i 3 21. -2-1 23.1-4 -1 3 2 3 113 22. 5-5-5 -1 4 2 3-5-3 0 5 1 0 001-2 002 1 24. 1-5 0 0 In Exercises 25 and 26, solve the linear system. 25. (1+i)x+iy = 5 + 4i (1 - i)x - 4y = −11+5i 26. (1)x (3+i)y=−5 − i (2+i)x+ (1+2i)y= 1+6i 4.7 Similarity Transformations and Diagonalization 325 Suppose that A is an (m x n) matrix and B is an (nx p) matrix. Use Exercise 36 and the properties of the transpose operation to give a quick proof that (AB)* = B* A*. 38. An (nx n) matrix A is called Hermitian if A* = A. a) Prove that a Hermitian matrix A has only real eigenvalues. [Hint: Observing that x¹x=x*I, modify the proof of Theorem 17.]
In Exercises 1-18, s = 1+2i, u = 3-2i, v = 4+i, w = 2-i, and z = 1+i. In each exercise, perform the indicated calculation and express the result in the form a + ib. 1. ū 4. Z + w 7. vv 10. z²w 13. u/v 16. (w + v)/u Find the eigenvalues and the eigenvectors for the matri- ces in Exercises 19-24. (For the matrix in Exercise 24, one eigenvalue is λ = 1 + 5i.) 19. 68 -1 2 27. x = 2. Z 5. u + ū 8. uv 11. ūw² 14. v/u² 17. w + iz 29. x = In Exercises 27-30, calculate ||x||. [ ] 1+i 2 1 3. u + v 6. s-s 9. s2 – w 12. s(u² + v) 15. s/z 18. s - iw 2i 3+i 20. 2⁰ [44] 28. x = 30. x = 3+i 2-i 2i 1-i 3 21. -2-1 23.1-4 -1 3 2 3 113 22. 5-5-5 -1 4 2 3-5-3 0 5 1 0 001-2 002 1 24. 1-5 0 0 In Exercises 25 and 26, solve the linear system. 25. (1+i)x+iy = 5 + 4i (1 - i)x - 4y = −11+5i 26. (1)x (3+i)y=−5 − i (2+i)x+ (1+2i)y= 1+6i 4.7 Similarity Transformations and Diagonalization 325 Suppose that A is an (m x n) matrix and B is an (nx p) matrix. Use Exercise 36 and the properties of the transpose operation to give a quick proof that (AB)* = B* A*. 38. An (nx n) matrix A is called Hermitian if A* = A. a) Prove that a Hermitian matrix A has only real eigenvalues. [Hint: Observing that x¹x=x*I, modify the proof of Theorem 17.]
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![In Exercises 1-18, s = 1+2i, u = 3 - 2i, v = 4+i,
w = 2-i, and z = 1+i. In each exercise, perform the
indicated calculation and express the result in the form
a + ib.
1. ū
4. Z + w
7. vv
10. z²w
13. u/v
16. (w + v)/u
2. Z
5. u + ū
8. uv
11. uw²
14. v/u²
17. w + iz
Find the eigenvalues and the eigenvectors for the matri-
ces in Exercises 19-24. (For the matrix in Exercise 24,
one eigenvalue is λ = 1 + 5i.)
19.
6 8
24
[49] 20. [44]
-1 2
27. x =
In Exercises 27-30, calculate ||x||.
1+i
2
3. u + v
6. s-s
9. s² - w
12. s(u² + v)
15. s/z
18. s - iw
1-2i
[B]
3+i
29. x =
28. x =
3+i
2-i
30. x =
2i
[4]
3
21.
[34]
23.
1 -4 -1
3 2 3
1 1 3
22.
24.
5-5-5
-1 4 2
3 -5 -3
1-5
0
0
5 1
0 0
1-2
0 0
00 2 1
In Exercises 25 and 26, solve the linear system.
25. (1 + i)x +iy = 5 + 4i
(1 - i)x - 4y = -11+5i
26. (1 - i)x (3+i)y=-5 - i
(2+i)x+ (1+2i)y= 1+6i
4.7 Similarity Transformations and Diagonalization
325
Suppose that A is an (m × n) matrix and B is an
(nx p) matrix. Use Exercise 36 and the properties
of the transpose operation to give a quick proof that
(AB)* = B* A*.
38. An (n xn) matrix A is called Hermitian if A* = A.
a) Prove that a Hermitian matrix A has only real
eigenvalues. [Hint: Observing that x¹ x = x*x,
modify the proof of Theorem 17.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7bf79e7e-db7e-4b5c-a300-38b09014f3cb%2F3b658eff-65f0-429a-9944-90a0a291b171%2Fu7ge0to_processed.jpeg&w=3840&q=75)
Transcribed Image Text:In Exercises 1-18, s = 1+2i, u = 3 - 2i, v = 4+i,
w = 2-i, and z = 1+i. In each exercise, perform the
indicated calculation and express the result in the form
a + ib.
1. ū
4. Z + w
7. vv
10. z²w
13. u/v
16. (w + v)/u
2. Z
5. u + ū
8. uv
11. uw²
14. v/u²
17. w + iz
Find the eigenvalues and the eigenvectors for the matri-
ces in Exercises 19-24. (For the matrix in Exercise 24,
one eigenvalue is λ = 1 + 5i.)
19.
6 8
24
[49] 20. [44]
-1 2
27. x =
In Exercises 27-30, calculate ||x||.
1+i
2
3. u + v
6. s-s
9. s² - w
12. s(u² + v)
15. s/z
18. s - iw
1-2i
[B]
3+i
29. x =
28. x =
3+i
2-i
30. x =
2i
[4]
3
21.
[34]
23.
1 -4 -1
3 2 3
1 1 3
22.
24.
5-5-5
-1 4 2
3 -5 -3
1-5
0
0
5 1
0 0
1-2
0 0
00 2 1
In Exercises 25 and 26, solve the linear system.
25. (1 + i)x +iy = 5 + 4i
(1 - i)x - 4y = -11+5i
26. (1 - i)x (3+i)y=-5 - i
(2+i)x+ (1+2i)y= 1+6i
4.7 Similarity Transformations and Diagonalization
325
Suppose that A is an (m × n) matrix and B is an
(nx p) matrix. Use Exercise 36 and the properties
of the transpose operation to give a quick proof that
(AB)* = B* A*.
38. An (n xn) matrix A is called Hermitian if A* = A.
a) Prove that a Hermitian matrix A has only real
eigenvalues. [Hint: Observing that x¹ x = x*x,
modify the proof of Theorem 17.]
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 4 steps with 78 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
