Lesson14: Eigenvalues And Eigenvectors

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Lesson14: Eigenvalues And Eigenvectors

  1. 1. Lesson 14 Eigenvalues and Eigenvectors Math 20 October 22, 2007 Announcements Midterm almost done Problem Set 5 is on the WS. Due October 24 OH: Mondays 1–2, Tuesdays 3–4, Wednesdays 1–3 (SC 323) Prob. Sess.: Sundays 6–7 (SC B-10), Tuesdays 1–2 (SC 116)
  2. 2. Geometric effect of a diagonal linear transformation Example 20 Let A = . Draw the effect of the linear transformation 0 −1 which is multiplication by A.
  3. 3. Geometric effect of a diagonal linear transformation Example 20 Let A = . Draw the effect of the linear transformation 0 −1 which is multiplication by A. y x
  4. 4. Geometric effect of a diagonal linear transformation Example 20 Let A = . Draw the effect of the linear transformation 0 −1 which is multiplication by A. y x e1
  5. 5. Geometric effect of a diagonal linear transformation Example 20 Let A = . Draw the effect of the linear transformation 0 −1 which is multiplication by A. y x e1 Ae1
  6. 6. Geometric effect of a diagonal linear transformation Example 20 Let A = . Draw the effect of the linear transformation 0 −1 which is multiplication by A. y e2 x e1 Ae1
  7. 7. Geometric effect of a diagonal linear transformation Example 20 Let A = . Draw the effect of the linear transformation 0 −1 which is multiplication by A. y e2 x e1 Ae1 Ae2
  8. 8. Geometric effect of a diagonal linear transformation Example 20 Let A = . Draw the effect of the linear transformation 0 −1 which is multiplication by A. y e2 v x e1 Ae1 Ae2
  9. 9. Geometric effect of a diagonal linear transformation Example 20 Let A = . Draw the effect of the linear transformation 0 −1 which is multiplication by A. y e2 v x e1 Ae1 Ae2
  10. 10. Geometric effect of a diagonal linear transformation Example 20 Let A = . Draw the effect of the linear transformation 0 −1 which is multiplication by A. y e2 v x e1 Ae1 Ae2 Av
  11. 11. Geometric effect of a diagonal linear transformation Example 20 Let A = . Draw the effect of the linear transformation 0 −1 which is multiplication by A. y e2 v x e1 Ae1 Ae2 Av
  12. 12. Geometric effect of a diagonal linear transformation Example 20 Let A = . Draw the effect of the linear transformation 0 −1 which is multiplication by A. y e2 v x e1 Ae1 Ae2 Av
  13. 13. Example Draw the effect of the linear transformation which is multiplication by A2 . y x
  14. 14. Example Draw the effect of the linear transformation which is multiplication by A2 . y x
  15. 15. Example Draw the effect of the linear transformation which is multiplication by A2 . y x
  16. 16. Geometric effect of a non-diagonal linear transformation Example 1/2 3/2 Let B = . Draw the effect of the linear transformation 3/2 1/2 which is multiplication by B. y x
  17. 17. Geometric effect of a non-diagonal linear transformation Example 1/2 3/2 Let B = . Draw the effect of the linear transformation 3/2 1/2 which is multiplication by B. y x e1
  18. 18. Geometric effect of a non-diagonal linear transformation Example 1/2 3/2 Let B = . Draw the effect of the linear transformation 3/2 1/2 which is multiplication by B. y Be1 x e1
  19. 19. Geometric effect of a non-diagonal linear transformation Example 1/2 3/2 Let B = . Draw the effect of the linear transformation 3/2 1/2 which is multiplication by B. y e2 Be1 x e1
  20. 20. Geometric effect of a non-diagonal linear transformation Example 1/2 3/2 Let B = . Draw the effect of the linear transformation 3/2 1/2 which is multiplication by B. y e2 Be1 Be2 x e1
  21. 21. Geometric effect of a non-diagonal linear transformation Example 1/2 3/2 Let B = . Draw the effect of the linear transformation 3/2 1/2 which is multiplication by B. y e2 Be1 v Be2 x e1
  22. 22. Geometric effect of a non-diagonal linear transformation Example 1/2 3/2 Let B = . Draw the effect of the linear transformation 3/2 1/2 which is multiplication by B. y e2 Be1 v Be2 x e1
  23. 23. Geometric effect of a non-diagonal linear transformation Example 1/2 3/2 Let B = . Draw the effect of the linear transformation 3/2 1/2 which is multiplication by B. y e2 Be1 v Be2 x e1 Bv
  24. 24. Geometric effect of a non-diagonal linear transformation Example 1/2 3/2 Let B = . Draw the effect of the linear transformation 3/2 1/2 which is multiplication by B. y e2 Be1 v Be2 x e1 Bv
  25. 25. Geometric effect of a non-diagonal linear transformation Example 1/2 3/2 Let B = . Draw the effect of the linear transformation 3/2 1/2 which is multiplication by B. y e2 Be1 v Be2 x e1 Bv
  26. 26. Geometric effect of a non-diagonal linear transformation Example 1/2 3/2 Let B = . Draw the effect of the linear transformation 3/2 1/2 which is multiplication by B. y x
  27. 27. Geometric effect of a non-diagonal linear transformation Example 1/2 3/2 Let B = . Draw the effect of the linear transformation 3/2 1/2 which is multiplication by B. y v1 x
  28. 28. Geometric effect of a non-diagonal linear transformation Example 1/2 3/2 Let B = . Draw the effect of the linear transformation 3/2 1/2 which is multiplication by B. y Bv1 v1 x
  29. 29. Geometric effect of a non-diagonal linear transformation Example 1/2 3/2 Let B = . Draw the effect of the linear transformation 3/2 1/2 which is multiplication by B. y Bv1 v1 x v2
  30. 30. Geometric effect of a non-diagonal linear transformation Example 1/2 3/2 Let B = . Draw the effect of the linear transformation 3/2 1/2 which is multiplication by B. y Bv1 Bv2 v1 x v2
  31. 31. Geometric effect of a non-diagonal linear transformation Example 1/2 3/2 Let B = . Draw the effect of the linear transformation 3/2 1/2 which is multiplication by B. y Bv1 Bv2 v1 x 2e2 v2
  32. 32. Geometric effect of a non-diagonal linear transformation Example 1/2 3/2 Let B = . Draw the effect of the linear transformation 3/2 1/2 which is multiplication by B. y Bv1 Bv2 v1 x 2e2 v2
  33. 33. Geometric effect of a non-diagonal linear transformation Example 1/2 3/2 Let B = . Draw the effect of the linear transformation 3/2 1/2 which is multiplication by B. y Bv1 Bv2 v1 x 2e2 v2
  34. 34. Geometric effect of a non-diagonal linear transformation Example 1/2 3/2 Let B = . Draw the effect of the linear transformation 3/2 1/2 which is multiplication by B. y 2Be2 Bv1 Bv2 v1 x 2e2 v2
  35. 35. Geometric effect of a non-diagonal linear transformation Example 1/2 3/2 Let B = . Draw the effect of the linear transformation 3/2 1/2 which is multiplication by B. y 2Be2 Bv1 Bv2 v1 x 2e2 v2
  36. 36. Geometric effect of a non-diagonal linear transformation Example 1/2 3/2 Let B = . Draw the effect of the linear transformation 3/2 1/2 which is multiplication by B. y 2Be2 Bv1 Bv2 v1 x 2e2 v2
  37. 37. Geometric effect of a non-diagonal linear transformation Example 1/2 3/2 Let B = . Draw the effect of the linear transformation 3/2 1/2 which is multiplication by B. y 2Be2 Bv1 Bv2 v1 x 2e2 v2
  38. 38. The big concept Definition Let A be an n × n matrix. The number λ is called an eigenvalue of A if there exists a nonzero vector x ∈ Rn such that Ax = λx. (1) Every nonzero vector satisfying (1) is called an eigenvector of A associated with the eigenvalue λ.
  39. 39. Finding the eigenvector(s) corresponding to an eigenvalue Example (Worksheet Problem 1) 0 −2 Let A = . The number 3 is an eigenvalue for A. Find −3 1 an eigenvector corresponding to this eigenvalue.
  40. 40. Finding the eigenvector(s) corresponding to an eigenvalue Example (Worksheet Problem 1) 0 −2 Let A = . The number 3 is an eigenvalue for A. Find −3 1 an eigenvector corresponding to this eigenvalue. Solution We seek x such that Ax = 3x =⇒ (A − 3I)x = 0.
  41. 41. Finding the eigenvector(s) corresponding to an eigenvalue Example (Worksheet Problem 1) 0 −2 Let A = . The number 3 is an eigenvalue for A. Find −3 1 an eigenvector corresponding to this eigenvalue. Solution We seek x such that Ax = 3x =⇒ (A − 3I)x = 0. −3 −2 0 1 2/3 0 A − 3I 0 = −3 −2 0 0 00
  42. 42. Finding the eigenvector(s) corresponding to an eigenvalue Example (Worksheet Problem 1) 0 −2 Let A = . The number 3 is an eigenvalue for A. Find −3 1 an eigenvector corresponding to this eigenvalue. Solution We seek x such that Ax = 3x =⇒ (A − 3I)x = 0. −3 −2 0 1 2/3 0 A − 3I 0 = −3 −2 0 0 00 −2/3 −2 So , or , are possible eigenvectors. 1 3
  43. 43. Example (Worksheet Problem 2) The number −2 is also an eigenvalue for A. Find an eigenvector.
  44. 44. Example (Worksheet Problem 2) The number −2 is also an eigenvalue for A. Find an eigenvector. Solution We have 2 −2 1 −1 A + 2I = . −3 3 00 1 is an eigenvector for the eigenvalue −2. So 1
  45. 45. Summary To find the eigenvector(s) of a matrix corresponding to an eigenvalue λ, do Gaussian Elimination on A − λI.
  46. 46. Find the eigenvalues of a matrix Example (Worksheet Problem 3) Determine the eigenvalues of −4 −3 A= . 3 6
  47. 47. Find the eigenvalues of a matrix Example (Worksheet Problem 3) Determine the eigenvalues of −4 −3 A= . 3 6 Solution Ax = λx has a nonzero solution if and only if (A − λI)x = 0 has a nonzero solution, if and only if A − λI is not invertible.
  48. 48. Find the eigenvalues of a matrix Example (Worksheet Problem 3) Determine the eigenvalues of −4 −3 A= . 3 6 Solution Ax = λx has a nonzero solution if and only if (A − λI)x = 0 has a nonzero solution, if and only if A − λI is not invertible. What magic number determines whether a matrix is invertible?
  49. 49. Find the eigenvalues of a matrix Example (Worksheet Problem 3) Determine the eigenvalues of −4 −3 A= . 3 6 Solution Ax = λx has a nonzero solution if and only if (A − λI)x = 0 has a nonzero solution, if and only if A − λI is not invertible. What magic number determines whether a matrix is invertible? The determinant!
  50. 50. Find the eigenvalues of a matrix Example (Worksheet Problem 3) Determine the eigenvalues of −4 −3 A= . 3 6 Solution Ax = λx has a nonzero solution if and only if (A − λI)x = 0 has a nonzero solution, if and only if A − λI is not invertible. What magic number determines whether a matrix is invertible? The determinant! So to find the possible values λ for which this could be true, we need to find the determinant of A − λI.
  51. 51. −4 − λ −3 det(A − λI) = 6−λ 3 = (−4 − λ)(6 − λ) − (−3)(3) = (−24 − 2λ + λ2 ) + 9 = λ2 − 2λ − 15 = (λ + 3)(λ − 5) So λ = −3 and λ = 5 are the eigenvalues for A.
  52. 52. We can find the eigenvectors now, based on what we did before. We have A + 3I =
  53. 53. We can find the eigenvectors now, based on what we did before. We have −1 −3 A + 3I = 3 9
  54. 54. We can find the eigenvectors now, based on what we did before. We have −1 −3 13 A + 3I = 3 9 00
  55. 55. We can find the eigenvectors now, based on what we did before. We have −1 −3 13 A + 3I = 3 9 00 −3 So is an eigenvector for this eigenvalue. 1
  56. 56. We can find the eigenvectors now, based on what we did before. We have −1 −3 13 A + 3I = 3 9 00 −3 So is an eigenvector for this eigenvalue. Also, 1 A − 5I =
  57. 57. We can find the eigenvectors now, based on what we did before. We have −1 −3 13 A + 3I = 3 9 00 −3 So is an eigenvector for this eigenvalue. Also, 1 −9 −3 A − 5I = 3 1
  58. 58. We can find the eigenvectors now, based on what we did before. We have −1 −3 13 A + 3I = 3 9 00 −3 So is an eigenvector for this eigenvalue. Also, 1 −9 −3 1 1/3 A − 5I = 3 1 0 0
  59. 59. We can find the eigenvectors now, based on what we did before. We have −1 −3 13 A + 3I = 3 9 00 −3 So is an eigenvector for this eigenvalue. Also, 1 −9 −3 1 1/3 A − 5I = 3 1 0 0 −1/3 −1 So or would be an eigenvector for this eigenvalue. 1 3
  60. 60. Summary To find the eigenvalues of a matrix A, find the determinant of A − λI. This will be a polynomial in λ, and its roots are the eigenvalues.
  61. 61. Example (Worksheet Problem 4) Find the eigenvalues of −10 6 A= . −15 9
  62. 62. Example (Worksheet Problem 4) Find the eigenvalues of −10 6 A= . −15 9 Solution The characteristic polynomial is −10 − λ 6 = (−10−λ)(9−λ)−(−15)(6) = λ2 +λ = λ(λ+1). −15 9−λ
  63. 63. Example (Worksheet Problem 4) Find the eigenvalues of −10 6 A= . −15 9 Solution The characteristic polynomial is −10 − λ 6 = (−10−λ)(9−λ)−(−15)(6) = λ2 +λ = λ(λ+1). −15 9−λ The eigenvalues are 0 and −1.
  64. 64. Example (Worksheet Problem 4) Find the eigenvalues of −10 6 A= . −15 9 Solution The characteristic polynomial is −10 − λ 6 = (−10−λ)(9−λ)−(−15)(6) = λ2 +λ = λ(λ+1). −15 9−λ 3 The eigenvalues are 0 and −1. A set of eigenvectors are and 5 2 . 3
  65. 65. Applications In a Markov Chain, the steady-state vector is an eigenvector corresponding to the eigenvalue 1.

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