theorem 5.79 Spectral theorem for a self-adjoint operator

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:3416 · p. 137

Rests on

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 equation 8.6 eq:cpx-contour-integral ¶
      - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
      - proves proof ch:06-complex-analysis@proof-5 ¶
    - depends_on theorem 8.17 Derivatives of all orders; Cauchy estimates ¶
      - depends_on lemma 8.9 ML estimate ¶
      - depends_on theorem 8.16 Cauchy integral formula ¶
      - proves proof ch:06-complex-analysis@proof-11 ¶
    - 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 definition 5.37 Linear transformation ¶
    - 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 definition 5.37 Linear transformation ¶ ↺
      - depends_on definition 5.7 Vector subspace ¶
      - proves proof ch:03-linear-algebra-representations@proof-12 ¶
  - 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 definition 5.47 Inverse of a linear transformation ¶
      - depends_on definition 5.37 Linear transformation ¶ ↺
      - proves proof ch:03-linear-algebra-representations@proof-17 ¶
    - depends_on theorem 5.40 Rank–nullity ¶
      - depends_on definition 5.15 Basis ¶
      - depends_on definition 5.39 Kernel, image, nullity, rank ¶ ↺
      - depends_on lemma 5.38 Exchange and completion ¶
      - proves proof ch:03-linear-algebra-representations@proof-13 ¶
    - 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 4.4 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 5.37 Linear transformation ¶ ↺
depends_on theorem 5.40 Rank–nullity ¶ ↺
proves proof ch:03-linear-algebra-representations@proof-31 ¶

Supports

depends_on definition 30.20 Pressure and deviatoric stress ¶
- depends_on definition 30.31 The engineering moduli ¶
  - depends_on phenomenon 30.70 Hertzian contact ¶
    - depends_on definition A.698 The Hertz contact problem ¶
      - depends_on lemma A.700 The displacement produced by an elliptic pressure ¶
      - depends_on proposition A.701 The two-body compliance rule ¶
      - depends_on remark A.699 The third condition is not decoration ¶
      - depends_on theorem A.702 Hertz's solution ¶
    - depends_on example A.703 The numbers of the chapter, recomputed ¶
    - depends_on example 30.71 Contact stresses are enormous ¶
  - depends_on phenomenon 30.60 Coulomb's law of torsion ¶
    - depends_on example 30.61 A torsion fibre as a force sensor ¶
    - depends_on phenomenon 34.3 The stiffness of a twisted wire is geometric ¶
    - depends_on remark 30.62 Saint-Venant's theory: non-circular sections warp ¶
    - tests experiment Coulomb: the torsion pendulum ¶
  - 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 30.51 Moment and curvature ¶
    - depends_on definition A.680 Kirchhoff kinematics ¶
      - depends_on lemma A.687 Kinetic energy ¶
      - depends_on lemma A.681 Strains under the Kirchhoff hypothesis ¶
    - depends_on definition 30.63 Flexural rigidity ¶ ↺
    - depends_on phenomenon 30.52 Euler buckling ¶
      - depends_on example 30.53 A steel rod buckles at a twentieth of its crushing load ¶
      - depends_on remark 30.54 Other end conditions, and what buckling is variationally ¶
      - depends_on remark 30.55 Beyond the critical load: the elastica ¶
      - depends_on remark 30.68 Shells carry load in a different way ¶ ↺
    - depends_on theorem 30.56 The beam equation ¶
      - depends_on example 30.58 A steel ruler ¶
      - depends_on proposition 30.57 Bending waves are dispersive ¶
  - depends_on remark A.676 The third cubic constant is measured, and it is not small ¶
  - depends_on theorem A.695 Boussinesq's point-force solution, quoted ¶
    - depends_on definition A.698 The Hertz contact problem ¶ ↺
    - depends_on lemma A.697 Surface displacement under a distributed pressure ¶
      - depends_on lemma A.700 The displacement produced by an elliptic pressure ¶ ↺
      - depends_on proposition A.701 The two-body compliance rule ¶ ↺
- depends_on definition 30.79 Maxwell viscoelasticity ¶
  - depends_on proposition 30.80 A Maxwell material is a solid or a fluid depending on how long you watch ¶
- depends_on proposition 30.74 The maximum shear stress, and its indifference to pressure ¶
  - depends_on phenomenon 30.73 Yield and brittle fracture ¶
  - depends_on remark A.705 The dangerous stress is below the surface ¶
  - depends_on remark 30.75 Two criteria, one uncited ¶
- depends_on remark 30.75 Two criteria, one uncited ¶ ↺
depends_on proposition 30.9 Principal strains ¶
- depends_on example 30.10 Simple shear and pure shear are the same strain ¶
depends_on theorem 5.80 Simultaneous diagonalization of commuting self-adjoint operators ¶
- depends_on corollary 5.82 Spectral theorem for a normal operator ¶
depends_on theorem 5.84 Spectral theorem for a real symmetric operator ¶
- 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 lemma A.610 Block positivity ¶
  - depends_on proposition A.611 The constraint on $\Gamma$ ¶
    - depends_on proposition A.612 The dictionary between $A$ and $\Gamma$ ¶
- depends_on lemma A.601 Several variables, complex symmetric matrix ¶
  - depends_on proposition A.612 The dictionary between $A$ and $\Gamma$ ¶ ↺
  - depends_on proposition A.602 The Wigner function of a Gaussian is a positive Gaussian ¶
    - depends_on proposition A.604 Non-negativity forbids a vanishing overlap ¶
      - depends_on proposition A.609 The exponent is a quadratic polynomial ¶
- depends_on lemma A.600 The real part of an inverse ¶
  - depends_on lemma A.601 Several variables, complex symmetric matrix ¶ ↺
  - depends_on proposition A.612 The dictionary between $A$ and $\Gamma$ ¶ ↺
- depends_on proposition 5.90 Polar decomposition ¶
- 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 proposition 5.92 The eigenvalues do not control the norm ¶ ↺
- 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 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 29.16 Principal axes ¶
- depends_on definition 29.19 Inertia ellipsoid ¶
  - depends_on theorem 29.29 Poinsot's construction ¶
    - depends_on remark 29.32 What the linearization is entitled to conclude ¶
    - depends_on remark 29.30 Polhode and herpolhode ¶
- depends_on definition 29.35 The Lagrange top ¶
- depends_on definition 29.18 Classification of tops ¶
  - depends_on proposition 29.28 Free rotation of a symmetric body ¶
    - depends_on phenomenon 29.34 Free nutation of the Earth ¶
- depends_on phenomenon 29.31 The intermediate-axis instability ¶
  - depends_on remark 29.32 What the linearization is entitled to conclude ¶ ↺
- depends_on proposition 29.33 Dissipation drives a free body to its largest moment ¶
- depends_on proposition 29.17 Triangle inequalities ¶
  - depends_on proposition 29.22 Perpendicular-axis theorem ¶
- depends_on theorem 29.25 Euler's equations ¶
  - depends_on phenomenon 29.46 A fast spinning top does not fall ¶
    - depends_on phenomenon 29.49 Precession of the equinoxes ¶
      - depends_on proposition 29.50 The principal nutation ¶
  - depends_on phenomenon 29.31 The intermediate-axis instability ¶ ↺
  - depends_on proposition 29.26 The two integrals of torque-free motion ¶
    - depends_on proposition 29.33 Dissipation drives a free body to its largest moment ¶ ↺
    - depends_on proposition 29.43 The three integrable heavy tops ¶
      - depends_on remark 29.45 What integrability buys, and where it stops ¶
      - depends_on remark 29.44 The Kovalevskaya integral ¶
    - depends_on remark 29.27 Euler's equations as a Lie–Poisson system ¶
    - depends_on remark 29.30 Polhode and herpolhode ¶ ↺
    - depends_on remark 24.42 The intermediate axis ¶
    - depends_on theorem 29.29 Poinsot's construction ¶ ↺
  - depends_on proposition 29.28 Free rotation of a symmetric body ¶ ↺
  - depends_on proposition 24.41 The free rigid body is a Lie–Poisson system ¶
    - depends_on remark 24.48 Reduction is what physicists do without saying so ¶
      - depends_on example A.573 The abelian case, and eliminating a cyclic coordinate ¶
    - depends_on remark 24.42 The intermediate axis ¶ ↺

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	Existence of an eigenvalue over $\C$	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3432
`depends_on`	→	Adjoint	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3432
`depends_on`	→	Rank–nullity	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3432
`depends_on`	←	Pressure and deviatoric stress	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:644
`depends_on`	←	Principal strains	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:273
`depends_on`	←	Simultaneous diagonalization of commuting self-adjoint operators	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3513
`depends_on`	←	Spectral theorem for a real symmetric operator	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3697
`depends_on`	←	Normal modes	declared	parts/03-classical-mechanics/11-oscillations-waves.tex:1127
`depends_on`	←	Principal axes	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:466
`proves`	←	ch:03-linear-algebra-representations@proof-31	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3436