theorem 28.33 Normal modes

open in the book · parts/03-classical-mechanics/11-oscillations-waves.tex:1105 · p. 957

Rests on

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 ¶
      - depends_on proposition 8.10 Fundamental theorem for contours ¶
      - depends_on theorem 8.17 Derivatives of all orders; Cauchy estimates ¶
      - proves proof ch:06-complex-analysis@proof-12 ¶
    - 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 5.56 Endomorphism ¶
      - depends_on equation 5.19 eq:lin-leibniz-det ¶
      - depends_on equation 5.74 eq:lin-matrix-rep ¶
      - proves proof ch:03-linear-algebra-representations@proof-25 ¶
    - depends_on definition 5.69 Eigenvector, eigenvalue, eigenspace ¶
      - depends_on definition 5.56 Endomorphism ¶ ↺
      - depends_on definition 5.39 Kernel, image, nullity, rank ¶
    - depends_on proposition 5.49 Injective, surjective, invertible ¶
      - depends_on definition 5.39 Kernel, image, nullity, rank ¶ ↺
      - depends_on proposition 5.48 Existence, uniqueness, and linearity of the inverse ¶
      - depends_on theorem 5.40 Rank–nullity ¶
      - proves proof ch:03-linear-algebra-representations@proof-18 ¶
    - 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 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.32 Field ¶
      - depends_on definition 4.31 Module ¶
  - depends_on definition 5.37 Linear transformation ¶
    - depends_on definition 4.33 Vector space ¶ ↺
- depends_on theorem 5.40 Rank–nullity ¶ ↺
- 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 ¶

Supports

depends_on example 28.39 The linear triatomic molecule ¶
depends_on proposition 28.40 Modes and dispersion of the loaded string ¶
depends_on remark 28.34 Part II owes the simultaneous diagonalization of two quadratic forms ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	eq:osc-secular	declared	parts/03-classical-mechanics/11-oscillations-waves.tex:1127
`depends_on`	→	Spectral theorem for a self-adjoint operator	declared	parts/03-classical-mechanics/11-oscillations-waves.tex:1127
`depends_on`	→	Sylvester's law of inertia	declared	parts/03-classical-mechanics/11-oscillations-waves.tex:1127
`depends_on`	←	The linear triatomic molecule	declared	parts/03-classical-mechanics/11-oscillations-waves.tex:1331
`depends_on`	←	Modes and dispersion of the loaded string	declared	parts/03-classical-mechanics/11-oscillations-waves.tex:1361
`depends_on`	←	Part II owes the simultaneous diagonalization of two quadratic forms	declared	parts/03-classical-mechanics/11-oscillations-waves.tex:1206
`proves`	←	ch:11-oscillations-waves@proof-17	declared	parts/03-classical-mechanics/11-oscillations-waves.tex:1130