Kreyzig ch 08 linear algebra

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Kreyzig ch 08 linear algebra

  1. 1. Matrix Eigenvalue Problems TF2101 Matematika Rekayasa Sistem
  2. 2. Advanced Engineering Mathematics by Erwin Kreyszig Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.
  3. 3. Advanced Engineering Mathematics by Erwin Kreyszig Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.
  4. 4. Advanced Engineering Mathematics by Erwin Kreyszig Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.
  5. 5. Advanced Engineering Mathematics by Erwin Kreyszig Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.
  6. 6. Advanced Engineering Mathematics by Erwin Kreyszig Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.
  7. 7. Advanced Engineering Mathematics by Erwin Kreyszig Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.
  8. 8. Advanced Engineering Mathematics by Erwin Kreyszig Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.
  9. 9. Advanced Engineering Mathematics by Erwin Kreyszig Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.
  10. 10. Advanced Engineering Mathematics by Erwin Kreyszig Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.
  11. 11. Matrix Eigenvalue Problems Advanced Engineering Mathematics by Erwin Kreyszig Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.
  12. 12. − 5 2  A=  2 − 2    x   − 5 2  x1     = λ  1  Ax =    x   2 − 2  x 2   2 − 5 x1 + 2 x 2 = λx1 2 x1 − 2 x 2 = λx 2 (−5 − λ ) x1 + 2 x 2 = 0 2 x1 + (−2 − λ ) x 2 = 0 Advanced Engineering Mathematics by Erwin Kreyszig Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.
  13. 13. Pages 335-336a Continued Advanced Engineering Mathematics by Erwin Kreyszig Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.
  14. 14. Pages 335-336b Advanced Engineering Mathematics by Erwin Kreyszig Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.
  15. 15. Advanced Engineering Mathematics by Erwin Kreyszig Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.
  16. 16. Page 337a Continued Advanced Engineering Mathematics by Erwin Kreyszig Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.
  17. 17. Page 337b Advanced Engineering Mathematics by Erwin Kreyszig Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.
  18. 18. Page 338 Advanced Engineering Mathematics by Erwin Kreyszig Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.
  19. 19. Pages 339-340a Continued Advanced Engineering Mathematics by Erwin Kreyszig Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.
  20. 20. Pages 339-340b Continued Advanced Engineering Mathematics by Erwin Kreyszig Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.
  21. 21. Pages 339-340c Advanced Engineering Mathematics by Erwin Kreyszig Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.
  22. 22. Advanced Engineering Mathematics by Erwin Kreyszig Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.
  23. 23. Advanced Engineering Mathematics by Erwin Kreyszig Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.
  24. 24. Page 346 (3) Advanced Engineering Mathematics by Erwin Kreyszig Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.
  25. 25. Advanced Engineering Mathematics by Erwin Kreyszig Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.
  26. 26. Eigenbases, Diagonalization, Quadratic Forms Advanced Engineering Mathematics by Erwin Kreyszig Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.
  27. 27. Advanced Engineering Mathematics by Erwin Kreyszig Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.
  28. 28. Example Advanced Engineering Mathematics by Erwin Kreyszig Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.
  29. 29. Advanced Engineering Mathematics by Erwin Kreyszig Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.
  30. 30. Page 352 >> [V,E] =eig(A) V= -0.3015 0.4364 -0.3015 0.3015 0.2182 0.9045 -0.9045 0.8729 -0.3015 E= -4.0000 0 0 0 -0.0000 0 0 0 3.0000 Advanced Engineering Mathematics by Erwin Kreyszig Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.
  31. 31. Quadratic Forms. Transformation to Principal Axes Advanced Engineering Mathematics by Erwin Kreyszig Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.
  32. 32. Page 353 (2) Advanced Engineering Mathematics by Erwin Kreyszig Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.
  33. 33. Page 353 (3) Advanced Engineering Mathematics by Erwin Kreyszig Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.
  34. 34. Page 354 (1) Advanced Engineering Mathematics by Erwin Kreyszig Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.
  35. 35. Page 354 (2a) Continued Advanced Engineering Mathematics by Erwin Kreyszig Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.
  36. 36. Page 354 (2b) Advanced Engineering Mathematics by Erwin Kreyszig Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.
  37. 37. Complex Matrices and Forms The three classes of real matrices have complex counterparts that are of practical interest in certain applications, mainly because of their spectra, for instance in quantum mechanics. To define these classes, we need the following standard. Example if then and 6   3 + 4i 1 − i   3 − 4i 1 + i   3 − 4i T , A =  , A =  A=  6   6   1 + i 2 + 5i   2 − 5i  2 + 5i      Advanced Engineering Mathematics by Erwin Kreyszig Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.
  38. 38. unitary 1   1 i 3  1 − 3i  2 + i  4  3i 2  , B =  , C =  2 A= 1 + 3i − 2 + i − i  1  7  1 3     i   2  2 Example Hermitian Skew-Hermitian Advanced Engineering Mathematics by Erwin Kreyszig Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.
  39. 39. unitary 1   1 3  i 1 − 3i  2 + i  4  3i 2 2  , B =  A= 1 + 3i   − 2 + i − i , C =  1  1  7      3 i   2  2 Example Hermitian Skew-Hermitian Characteristic equation A : λ2 − 11λ + 18 = 0 B : λ2 − 2iλ + 8 = 0 C: λ 2 − iλ − 1 = 0 Eigenvalues 1 2 ( 9, 2 4i, − 2i 1 3+i , − 3+i 2 ) ( ) Advanced Engineering Mathematics by Erwin Kreyszig Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.
  40. 40. Advanced Engineering Mathematics by Erwin Kreyszig Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.
  41. 41. Advanced Engineering Mathematics by Erwin Kreyszig Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.
  42. 42. Page 363a Continued Advanced Engineering Mathematics by Erwin Kreyszig Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.
  43. 43. Page 363b Advanced Engineering Mathematics by Erwin Kreyszig Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.

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