+ü - w u² + v) z - iw for the matri 23. 1-4-1 3 2 3 13 24. 3 0 Ï 1-5 0 51 00 0 1-2 0021 0 In Exercises 25 and 26, solve the linear system.
+ü - w u² + v) z - iw for the matri 23. 1-4-1 3 2 3 13 24. 3 0 Ï 1-5 0 51 00 0 1-2 0021 0 In Exercises 25 and 26, solve the linear system.
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
100%
![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%2F7a0fc5ca-6033-4e33-92e2-d62448664519%2Fjdo2kz_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.]
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