the reduction to hamiltonian schur form explained
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The Reduction to Hamiltonian Schur Form Explained David S. Watkins - PowerPoint PPT Presentation

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The Reduction to Hamiltonian Schur Form Explained David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University July 2006 p.1 Linear-Quadratic Gaussian Problem July 2006 p.2 Linear-Quadratic Gaussian

  1. Construction of Q T   ∗ ⇒ ∗       ∗ ⇒     0     ⇒ ∗     ⇒ ∗ July 2006 – p.10

  2. Construction of Q T   ∗ ⇒ ∗       ∗ ⇒     0     ⇒  0    ⇒ ∗ July 2006 – p.10

  3. Construction of Q T   ∗ ∗       ∗ ⇒     0      0    ⇒ ∗ July 2006 – p.10

  4. Construction of Q T   ∗ ∗       ∗ ⇒     0      0    ⇒ 0 July 2006 – p.10

  5. Construction of Q T   ∗ ⇒ ∗       ∗ ⇒     0     ⇒  0    ⇒ 0 July 2006 – p.10

  6. Construction of Q T   ∗ ⇒ ∗       ⇒ 0     0     ⇒  0    ⇒ 0 July 2006 – p.10

  7. Construction of Q T   ⇒ ∗ ⇒ ∗       0     ⇒ 0     ⇒  0    0 July 2006 – p.10

  8. Construction of Q T   ⇒ ∗ ⇒ 0       0     ⇒ 0     ⇒  0    0 July 2006 – p.10

  9. Construction of Q T   ⇒ ∗ ⇒ 0       0     ⇒ 0     ⇒  0    0 Done! July 2006 – p.10

  10. Why it Works July 2006 – p.11

  11. Why it Works H 2 stays in skew-Hamiltonian Schur form . . . July 2006 – p.11

  12. Why it Works H 2 stays in skew-Hamiltonian Schur form . . . . . . every step of the way. July 2006 – p.11

  13. Why it Works H 2 stays in skew-Hamiltonian Schur form . . . . . . every step of the way. H x = λ 1 x July 2006 – p.11

  14. Why it Works H 2 stays in skew-Hamiltonian Schur form . . . . . . every step of the way. H x = λ 1 x H 2 x = λ 2 1 x July 2006 – p.11

  15. Why it Works H 2 stays in skew-Hamiltonian Schur form . . . . . . every step of the way. H x = λ 1 x H 2 x = λ 2 1 x relationships preserved throughout the transformation July 2006 – p.11

  16. Close up July 2006 – p.12

  17. Close up H 2 x = xλ 2 1 July 2006 – p.12

  18. Close up H 2 x = xλ 2 1       λ 2 ∗ ∗ ∗ ∗ ∗ ∗ 0 1 λ 2 ∗ ∗ ∗ ∗ ∗ 0       2       λ 2       ∗ ∗ ∗ ∗ 0 λ 2  3      =       1 λ 2 ∗ ∗       1       λ 2 ∗ ∗ ∗       2       λ 2 ∗ ∗ ∗ ∗ 3 July 2006 – p.12

  19. Close up H 2 x = xλ 2 1       λ 2 ∗ ∗ ∗ ∗ ∗ ∗ 0 1 λ 2 ∗ ∗ ∗ ∗ ∗ 0       2       λ 2       ∗ ∗ ∗ ∗ 0 λ 2  3      =       1 λ 2 ∗ ∗       1       λ 2 ∗ ∗ ∗       2       λ 2 ∗ ∗ ∗ ∗ 3 July 2006 – p.12

  20.       λ 2 ∗ ∗ 1  λ 2 λ 2 ∗ ∗ ∗  =       1 2     λ 2 ∗ ∗ ∗ ∗ 3 July 2006 – p.13

  21.       λ 2 ⇒ ∗ ∗ 1  λ 2 λ 2 ⇒ ∗ ∗ ∗  =       1 2     λ 2 ∗ ∗ ∗ ∗ 3 July 2006 – p.13

  22.       ⇒ ∗ ∗ 0 0  λ 2 ⇒ ∗ ∗ ∗ ∗  =       1     λ 2 ∗ ∗ ∗ ∗ 3 July 2006 – p.13

  23.       ⇒ ∗ 0 0 0  λ 2 ⇒ ∗ ∗ ∗ ∗  =       1     λ 2 ∗ ∗ ∗ ∗ 3 July 2006 – p.13

  24.       ⇒ ∗ 0 0 0  λ 2 λ 2 ⇒ ∗ ∗ ∗  =       1 1     λ 2 ∗ ∗ ∗ ∗ 3 July 2006 – p.13

  25.       λ 2 ⇒ 0 0 0 2  λ 2 λ 2 ⇒ ∗ ∗ ∗  =       1 1     λ 2 ∗ ∗ ∗ ∗ 3 July 2006 – p.13

  26.       λ 2 0 0 2  λ 2 λ 2 ⇒ ∗ ∗ ∗  =       1 1     λ 2 ⇒ ∗ ∗ ∗ ∗ 3 July 2006 – p.13

  27.       λ 2 0 0 2  λ 2 ⇒ ∗ ∗ ∗  = 0 0       1     ⇒ ∗ ∗ ∗ ∗ ∗ July 2006 – p.13

  28.       λ 2 0 0 2  λ 2 ⇒ ∗ ∗  = 0 0 0       1     ⇒ ∗ ∗ ∗ ∗ ∗ July 2006 – p.13

  29.       λ 2 0 0 2  λ 2 ⇒ ∗ ∗  = 0 0 0       1     λ 2 ⇒ ∗ ∗ ∗ ∗ 1 July 2006 – p.13

  30.       λ 2 0 0 2  λ 2 λ 2 ⇒ ∗  = 0 0 0       1 3     λ 2 ⇒ ∗ ∗ ∗ ∗ 1 July 2006 – p.13

  31.       λ 2 0 0 2  λ 2 λ 2 ∗  = 0 0       1 3     λ 2 ∗ ∗ ∗ ∗ 1 July 2006 – p.13

  32.       λ 2 ∗ ∗ ∗ ∗ ∗ ∗ 0 2 λ 2 ∗ ∗ ∗ ∗ ∗ 0       3       λ 2       ∗ ∗ ∗ ∗ 0 1 λ 2       =       1 λ 2 0 0       2       λ 2 ∗    0   0  3       λ 2 ∗ ∗ ∗ ∗ 1 July 2006 – p.14

  33.       λ 2 ∗ ∗ ∗ ∗ ∗ ∗ 0 2 λ 2 ∗ ∗ ∗ ∗ ∗ 0       3       λ 2       ⇒ ∗ ∗ ∗ ∗ 0 1 λ 2       =       1 λ 2 0 0       2       λ 2 ∗    0   0  3       λ 2 ⇒ ∗ ∗ ∗ ∗ 1 July 2006 – p.14

  34.       λ 2 ∗ ∗ ∗ ∗ ∗ ∗ 0 2 λ 2 ∗ ∗ ∗ ∗ ∗ 0       3       λ 2       ⇒ ∗ ∗ ∗ ∗ 0 1 λ 2       =       1 λ 2 0 0       2       λ 2 ∗    0   0  3       λ 2 ⇒ ∗ ∗ ∗ ∗ 0 1 July 2006 – p.14

  35.       λ 2 ∗ ∗ ∗ ∗ ∗ ∗ 0 2 λ 2 ∗ ∗ ∗ ∗ ∗ 0       3       λ 2       ⇒ ∗ ∗ ∗ ∗ 0 1 λ 2       =       1 λ 2 0 0       2       λ 2 ∗    0   0  3       λ 2 ⇒ ∗ ∗ 0 0 0 1 July 2006 – p.14

  36.       λ 2 ∗ ∗ ∗ ∗ ∗ ∗ 0 2 λ 2 ⇒ ∗ ∗ ∗ ∗ ∗ 0       3       λ 2       ⇒ ∗ ∗ ∗ ∗ 0 1 λ 2       =       1 λ 2 0 0       2       λ 2 ⇒ ∗    0   0  3       λ 2 ⇒ ∗ ∗ 0 0 1 July 2006 – p.14

  37.       λ 2 ∗ ∗ ∗ ∗ ∗ ∗ 0 2 λ 2 ⇒ ∗ ∗ ∗ ∗ ∗ 0       3       λ 2       ⇒ ∗ ∗ ∗ ∗ ∗ 0 1 λ 2       =       1 λ 2 0 0       2       λ 2 ⇒ ∗ ∗    0   0  3       λ 2 ⇒ ∗ ∗ 0 0 1 July 2006 – p.14

  38.       λ 2 ∗ ∗ ∗ ∗ ∗ ∗ 0 2 λ 2 ⇒ ∗ ∗ ∗ ∗ ∗ 0       1       λ 2       ⇒ ∗ ∗ 0 0 0 3 λ 2       =       1 λ 2 0 0       2       λ 2 ⇒ ∗    0   0  1       λ 2 ⇒ ∗ ∗ 0 0 3 July 2006 – p.14

  39.       λ 2 ⇒ ∗ ∗ ∗ ∗ ∗ ∗ 0 2 λ 2 ⇒ ∗ ∗ ∗ ∗ ∗ 0       1       λ 2       ∗ ∗ 0 0 0 3 λ 2       =       1 λ 2 ⇒ 0 0       2       λ 2 ⇒ ∗    0   0  1       λ 2 ∗ ∗ 0 0 3 July 2006 – p.14

  40.       λ 2 ⇒ ∗ ∗ ∗ ∗ ∗ ∗ 0 2 λ 2 ⇒ ∗ ∗ ∗ ∗ ∗ ∗ 0       1       λ 2       ∗ ∗ 0 0 0 3 λ 2       =       1 λ 2 ⇒ ∗ 0 0       2       λ 2 ⇒ ∗    0   0  1       λ 2 ∗ ∗ 0 0 3 July 2006 – p.14

  41.       λ 2 ⇒ ∗ ∗ ∗ ∗ ∗ ∗ 0 1 λ 2 ⇒ ∗ ∗ ∗ 0 0 0       2       λ 2       ∗ ∗ 0 0 0 3 λ 2       =       1 λ 2 ⇒ 0 0       1       λ 2 ⇒ ∗    0   0  2       λ 2 ∗ ∗ 0 0 3 July 2006 – p.14

  42.       λ 2 ∗ ∗ ∗ ∗ ∗ ∗ 0 1 λ 2 ∗ ∗ ∗ 0 0 0       2       λ 2       ∗ ∗ 0 0 0 3 λ 2       =       1 λ 2 0 0       1       λ 2 ∗    0   0  2       λ 2 ∗ ∗ 0 0 3 July 2006 – p.14

  43.       λ 2 ∗ ∗ ∗ ∗ ∗ ∗ 0 1 λ 2 ∗ ∗ ∗ 0 0 0       2       λ 2       ∗ ∗ 0 0 0 3 λ 2       =       1 λ 2 0 0       1       λ 2 ∗    0   0  2       λ 2 ∗ ∗ 0 0 3 It looks like we’re back where we started, July 2006 – p.15

  44.       λ 2 ∗ ∗ ∗ ∗ ∗ ∗ 0 1 λ 2 ∗ ∗ ∗ 0 0 0       2       λ 2       ∗ ∗ 0 0 0 3 λ 2       =       1 λ 2 0 0       1       λ 2 ∗    0   0  2       λ 2 ∗ ∗ 0 0 3 It looks like we’re back where we started, but . . . July 2006 – p.15

  45.   ∗ ∗ ∗ ∗ ∗ λ 1 ∗ ∗ ∗ ∗ ∗       ∗ ∗ ∗ ∗ ∗   H =   − λ 1     ∗ ∗ ∗ ∗ ∗     ∗ ∗ ∗ ∗ ∗ July 2006 – p.16

  46.   ∗ ∗ ∗ ∗ ∗ λ 1 ∗ ∗ ∗ ∗ ∗       ∗ ∗ ∗ ∗ ∗   H =   − λ 1     ∗ ∗ ∗ ∗ ∗     ∗ ∗ ∗ ∗ ∗ Now we can deflate. July 2006 – p.16

  47. In Conclusion July 2006 – p.17

  48. In Conclusion By keeping H 2 in skew-Hamiltonian Schur form July 2006 – p.17

  49. In Conclusion By keeping H 2 in skew-Hamiltonian Schur form every step of the way July 2006 – p.17

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