proposition 28.40 Modes and dispersion of the loaded string

Rests on

• depends_on equation 28.38 eq:osc-small-eom ¶
• depends_on theorem 28.33 Normal modes ¶
  • depends_on equation 28.39 eq:osc-secular ¶
    • step_from equation 28.37 eq:osc-small-forms ¶
  • depends_on theorem 5.79 Spectral theorem for a self-adjoint operator ¶
    • depends_on corollary 5.72 Existence of an eigenvalue over $\C$ ¶
      • depends_on theorem 8.19 Fundamental theorem of algebra ¶
        • depends_on theorem 8.18 Liouville ¶
        • proves proof ch:06-complex-analysis@proof-13 ¶
      • depends_on theorem 5.71 The eigenvalues are the roots of the characteristic polynomial ¶
        • depends_on definition 5.70 Characteristic polynomial ¶
        • depends_on definition 5.69 Eigenvector, eigenvalue, eigenspace ¶
        • depends_on proposition 5.49 Injective, surjective, invertible ¶
        • proves proof ch:03-linear-algebra-representations@proof-26 ¶
      • proves proof ch:03-linear-algebra-representations@proof-27 ¶
    • depends_on definition 5.41 Adjoint ¶
      • depends_on definition 5.18 Inner product ¶
        • depends_on definition 4.33 Vector space ¶
      • depends_on definition 5.37 Linear transformation ¶
        • depends_on definition 4.33 Vector space ¶ ↺
    • depends_on theorem 5.40 Rank–nullity ¶
      • depends_on definition 5.15 Basis ¶
        • depends_on definition 5.12 Subspace generated by a set of vectors ¶
        • depends_on definition 5.14 Linear independence ¶
      • depends_on definition 5.39 Kernel, image, nullity, rank ¶
        • depends_on definition 5.37 Linear transformation ¶ ↺
        • depends_on definition 5.7 Vector subspace ¶
        • proves proof ch:03-linear-algebra-representations@proof-12 ¶
      • depends_on lemma 5.38 Exchange and completion ¶
        • depends_on definition 5.15 Basis ¶ ↺
        • depends_on definition 5.12 Subspace generated by a set of vectors ¶ ↺
        • depends_on definition 5.14 Linear independence ¶ ↺
        • proves proof ch:03-linear-algebra-representations@proof-11 ¶
      • proves proof ch:03-linear-algebra-representations@proof-13 ¶
    • proves proof ch:03-linear-algebra-representations@proof-31 ¶
  • depends_on theorem 5.116 Sylvester's law of inertia ¶
    • depends_on definition 5.110 Bilinear map ¶
      • depends_on definition 5.37 Linear transformation ¶ ↺
    • depends_on definition 5.111 Symmetric and antisymmetric forms ¶
      • depends_on definition 5.110 Bilinear map ¶ ↺
      • depends_on equation 5.137 eq:lin-bilinear-matrix ¶
    • depends_on equation 5.139 eq:lin-bilinear-congruence ¶
    • proves proof ch:03-linear-algebra-representations@proof-53 ¶
  • proves proof ch:11-oscillations-waves@proof-17 ¶
• proves proof ch:11-oscillations-waves@proof-20 ¶

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