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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.]

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.1: Real Numbers
Problem 30E
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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.]
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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