theorem 5.116 Sylvester's law of inertia

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:4989 · p. 154

Rests on

depends_on definition 5.110 Bilinear map ¶
- depends_on definition 5.37 Linear transformation ¶
  - depends_on definition 4.33 Vector space ¶
    - depends_on definition 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.4 Internal binary operation; magma ¶
    - depends_on definition 4.32 Field ¶
      - depends_on definition 4.11 Inverse element ¶
      - depends_on definition 4.20 Monoid ¶
      - depends_on definition 4.30 Ring ¶
    - depends_on definition 4.31 Module ¶
      - depends_on definition 4.29 Action ¶
      - depends_on definition 4.28 Distributivity ¶
      - depends_on definition 4.21 Group ¶
      - depends_on definition 4.30 Ring ¶ ↺
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 ¶

Supports

depends_on definition 5.117 Signature ¶
- depends_on proposition 5.94 Principal axes of a real quadratic form ¶
  - depends_on proposition 5.95 The focal polar equation of a conic ¶
depends_on definition 10.6 Type of a second-order operator ¶
- depends_on definition 10.79 Parabolic boundary ¶
  - depends_on theorem 10.80 Parabolic maximum principle ¶
    - depends_on corollary 10.81 Uniqueness for the heat equation ¶
    - depends_on proposition A.163 The viscous problem produces the entropy inequality ¶
- depends_on example 10.9 ex:pde-three-types ¶
- depends_on proposition 10.25 The backward heat problem is ill posed ¶
- depends_on theorem A.83 Local canonical form for an elliptic operator with Hölder coefficients ¶
- depends_on theorem 10.8 The type is a coordinate invariant ¶
  - depends_on lemma A.90 Affine normalization ¶
    - depends_on lemma A.91 Smallness after rescaling ¶
  - depends_on proposition 10.13 Canonical form in two variables ¶
    - depends_on theorem A.83 Local canonical form for an elliptic operator with Hölder coefficients ¶ ↺
    - depends_on theorem 10.55 d'Alembert's formula ¶
      - depends_on corollary 10.56 Domain of dependence and influence ¶
      - depends_on theorem 10.59 Kirchhoff's formula ¶
  - depends_on proposition 10.11 What a characteristic surface is ¶
    - depends_on corollary 10.12 Characteristics of the three canonical operators ¶
      - depends_on corollary 10.60 Huygens' principle in three space dimensions ¶
    - depends_on definition A.102 Normal Cauchy problem of order $k$ ¶
      - depends_on definition A.103 First-order quasilinear system with zero data ¶
      - depends_on lemma A.105 Formal reduction of a normal problem ¶
    - depends_on theorem 10.22 Cauchy–Kovalevskaya ¶
depends_on proposition 30.30 Stability bounds ¶
- depends_on phenomenon 30.43 Two bulk wave speeds ¶
  - depends_on phenomenon 30.47 The three seismic phases ¶
  - depends_on remark 30.46 Why the plane-wave route, and not the potentials ¶
- depends_on phenomenon 30.32 Poisson contraction ¶
  - depends_on proposition 30.33 Conversion among the moduli ¶
    - depends_on definition 30.63 Flexural rigidity ¶
      - depends_on proposition A.684 The plate energy ¶
      - depends_on proposition A.682 The plane-stress law ¶
      - depends_on proposition 30.66 Chladni scaling ¶
      - depends_on remark A.704 $E^{*}$ in two roles, and one trap ¶
      - depends_on remark A.686 Why the operator comes out as $\nabla^{4}$ ¶
      - depends_on remark A.683 Where the factor $1-\nu^{2}$ comes from ¶
      - depends_on remark 30.68 Shells carry load in a different way ¶
      - depends_on theorem 30.64 The Kirchhoff plate equation ¶
    - depends_on example 30.35 From single crystals to the isotropic aggregate ¶
    - depends_on proposition A.682 The plane-stress law ¶ ↺
    - depends_on proposition 30.44 The speed ratio is fixed by Poisson's ratio ¶
      - depends_on example 30.45 Elastic waves in steel, and the distance to a source ¶
    - depends_on remark 30.36 How the moduli are actually measured ¶
- depends_on proposition 30.34 Minimum of the potential energy ¶
  - depends_on example 30.35 From single crystals to the isotropic aggregate ¶ ↺
- depends_on theorem 30.42 Uniqueness in elastodynamics ¶
depends_on proposition 5.94 Principal axes of a real quadratic form ¶ ↺
depends_on remark 7.107 What the Hessian is for ¶
depends_on theorem 5.86 Simultaneous diagonalization of a definite pencil ¶
- depends_on corollary 5.89 Rayleigh–Ritz ¶
- depends_on proposition 5.88 Rayleigh quotient of a definite pencil ¶
  - depends_on corollary 5.91 The operator norm of a real array ¶
    - depends_on proposition 5.92 The eigenvalues do not control the norm ¶
  - depends_on corollary 5.89 Rayleigh–Ritz ¶ ↺
depends_on theorem 28.33 Normal modes ¶
- 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 ¶
depends_on theorem 10.8 The type is a coordinate invariant ¶ ↺

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`	→	Bilinear map	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4998
`depends_on`	→	Symmetric and antisymmetric forms	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4998
`depends_on`	→	eq:lin-bilinear-congruence	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4998
`depends_on`	←	Signature	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:5056
`depends_on`	←	Type of a second-order operator	declared	parts/02-mathematical-methods/08-pdes.tex:160
`depends_on`	←	Stability bounds	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:928
`depends_on`	←	Principal axes of a real quadratic form	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4107
`depends_on`	←	What the Hessian is for	declared	parts/02-mathematical-methods/05-real-analysis.tex:3255
`depends_on`	←	Simultaneous diagonalization of a definite pencil	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3781
`depends_on`	←	Normal modes	declared	parts/03-classical-mechanics/11-oscillations-waves.tex:1127
`depends_on`	←	The type is a coordinate invariant	declared	parts/02-mathematical-methods/08-pdes.tex:190
`proves`	←	ch:03-linear-algebra-representations@proof-53	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:5001