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Exploring Finite-Dimensional Vector Spaces: Spectral Theory, Eigenvalues, and Applications

This comprehensive study delves into finite-dimensional vector spaces, covering topics such as linear vector spaces, spectral theory for matrices, and the geometrical significance of eigenvalues. It explores the Fredholm Alternative Theorem, least squares solutions, and the Generalized Pencil-of-function Method. The discussion extends to bases, inner products, magnitude and direction, Gram-Schmidt orthogonalization, coordinate transformation, eigenpairs, invariant manifolds, and the Spectral Decomposition Theorem. With graphic explanations, it touches on the Maximum Principle, Courant Minimax Principle, Sturm Sequence, Singular Value Decomposition, and Extraction of Modal Coefficients.

robertgarrett
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Exploring Finite-Dimensional Vector Spaces: Spectral Theory, Eigenvalues, and Applications

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  1. Finite Dimensional Vector Spaces • Linear Vector Spaces • Spectral Theory for Matrices • Geometrical Significance of Eigenvalues • Fredholm Alternative Theorem • Least Squares Solutions – Pseudo Inverses • Generalized Pencil-of-Function Method

  2. Linear Vector Spaces

  3. Bases

  4. Inner Product

  5. Generalized Magnitude & Direction

  6. Gram-Schmidt Orthogonalization

  7. Coordinate Transformation

  8. Eigenpairs

  9. Invariant Manifold

  10. Invariant Manifold (cont’)

  11. Spectral Decomposition Theorem

  12. Graphic Explanation

  13. Maximum Principle

  14. Courant Minimax Principle

  15. Applications of Sturm Sequence

  16. Fredholm Alternative Theorem

  17. Fredholm Alternative Theorem(cont’)

  18. Least Squares Solutions

  19. Least Squares Pseudo Inverse

  20. Pseudo Inverse (cont’)

  21. Singular Value Decomposition

  22. Singular Value Decomposition(cont’)

  23. Singular Value Decomposition(cont’)

  24. Singular Value Decomposition(cont’)

  25. Generalized Pencil-of-function Method

  26. Generalized Pencil-of-function Method(cont’)

  27. Extraction of Modal Coefficients

  28. Choice of Parameter L

  29. Numerical Results 2m I(t) Einc(t) Gaussian pulse load N=114 By GPOF (L = N/2 = 57)  d = 9.3, 7.7, 0.45, 0.42, 0.057, 0.056, 0.039, 0.0388, 0.0052, .. z = -0.0204j0.281, -0.0266j0.642, -0.0046j1.17, 0.0039j1.27(?) b = 0.3835, 0.02456, 0.0009379, 0.0007506 f = 0.0694, 0.204, 0.372, 0.404(?) GHz f = 0.0684, 0.203, 0.391, 0.414GHz

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