definition 5.41 Adjoint

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:1863 · p. 118

ground object -- no derivation owed

Rests on

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 4.3 Algebraic structure ¶
      - depends_on definition 4.2 Binary operation ¶
  - depends_on definition 4.32 Field ¶
    - depends_on definition 4.11 Inverse element ¶
      - depends_on definition 4.9 Neutral element ¶
    - depends_on definition 4.20 Monoid ¶
      - depends_on definition 4.9 Neutral element ¶ ↺
      - depends_on definition 4.18 Semigroup ¶
    - depends_on definition 4.30 Ring ¶
      - depends_on definition 4.8 Commutativity; abelian structure ¶ ↺
      - depends_on definition 4.28 Distributivity ¶
      - depends_on definition 4.21 Group ¶
      - depends_on definition 4.18 Semigroup ¶ ↺
  - depends_on definition 4.31 Module ¶
    - depends_on definition 4.29 Action ¶
      - depends_on definition 4.9 Neutral element ¶ ↺
    - 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 definition 4.33 Vector space ¶ ↺

Supports

depends_on proposition 5.42 The adjoint exists, is unique, and is linear ¶
- depends_on corollary 5.82 Spectral theorem for a normal operator ¶
depends_on theorem 12.38 Existence and uniqueness of the adjoint ¶
- depends_on definition 12.41 The operator classes ¶
  - depends_on definition 12.90 Self-adjoint family; commutant; irreducibility ¶
    - depends_on proposition A.589 Cyclic subspaces and the rank of the average ¶
      - depends_on lemma A.590 The Gram matrix is universal ¶
      - depends_on proposition A.591 Any two irreducible Weyl systems are equivalent ¶
    - depends_on theorem A.579 Stone–von Neumann ¶
    - depends_on theorem 12.91 Schur's lemma, commutant form ¶
      - depends_on proposition A.591 Any two irreducible Weyl systems are equivalent ¶ ↺
      - depends_on theorem 12.114 Stone–von Neumann ¶
    - depends_on theorem 12.114 Stone–von Neumann ¶ ↺
  - depends_on definition 12.58 Projection-valued measure ¶
    - depends_on definition 12.60 Functional calculus ¶
      - depends_on proposition 12.61 Uniqueness of the continuous functional calculus ¶
    - depends_on lemma A.247 Integration against a projection-valued measure ¶
    - depends_on theorem A.238 Spectral theorem, both forms ¶
      - depends_on proposition A.280 Direct-integral form of the spectral theorem ¶
      - depends_on proposition A.261 Spectral theorem for a unitary operator ¶
      - depends_on theorem A.279 Gelfand–Maurin ¶
      - depends_on theorem A.253 Stone ¶
    - depends_on theorem 12.59 Spectral theorem for a bounded self-adjoint operator ¶
      - depends_on definition 12.60 Functional calculus ¶ ↺
      - depends_on theorem 12.107 Nuclear spectral theorem ¶
      - depends_on theorem 12.91 Schur's lemma, commutant form ¶ ↺
      - depends_on theorem 12.66 Stone ¶
  - depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶
    - depends_on definition A.580 Weyl operator ¶
      - depends_on definition A.585 The Gaussian average of the Weyl operators ¶
      - depends_on lemma A.581 Composition law ¶
    - depends_on definition 12.109 Weyl system ¶
      - depends_on corollary 12.112 Commutator of the momentum with a function of the position ¶
      - depends_on definition A.580 Weyl operator ¶ ↺
      - depends_on example 12.110 The Schrödinger system ¶
      - depends_on proposition 12.111 The Weyl relation is a covariance statement ¶
      - depends_on theorem A.579 Stone–von Neumann ¶ ↺
      - depends_on theorem 12.114 Stone–von Neumann ¶ ↺
    - depends_on lemma A.255 Smoothed vectors lie in the domain ¶
      - depends_on proposition A.256 Density ¶
    - depends_on lemma A.581 Composition law ¶ ↺
    - depends_on lemma A.582 Joint strong continuity ¶
      - depends_on lemma A.583 Absolutely convergent operator-valued integrals ¶
    - depends_on proposition A.256 Density ¶ ↺
    - depends_on proposition 12.65 Exponential of a bounded self-adjoint operator ¶
      - depends_on theorem 12.66 Stone ¶ ↺
    - depends_on proposition 12.67 The generator is symmetric, and generates the motion ¶
      - depends_on lemma A.257 Integrated form of the equation of motion ¶
      - depends_on proposition A.263 The two constructions are inverse ¶
      - depends_on proposition A.259 The generator is self-adjoint ¶
      - depends_on proposition 12.111 The Weyl relation is a covariance statement ¶ ↺
      - depends_on theorem A.253 Stone ¶ ↺
    - depends_on theorem A.253 Stone ¶ ↺
    - depends_on theorem 12.66 Stone ¶ ↺
    - depends_on theorem 25.34 Stone–von Neumann ¶
      - depends_on lemma 25.39 The low-degree images are forced ¶
      - depends_on theorem 25.38 Groenewold–van Hove ¶
  - depends_on lemma A.231 Restriction to an invariant closed subspace ¶
    - depends_on lemma A.233 Construction of the system ¶
      - depends_on lemma A.234 The eigenvalues tend to zero, with finite multiplicity ¶
  - depends_on lemma A.230 Sequential characterisation ¶
    - depends_on lemma A.232 Attainment ¶
      - depends_on lemma A.233 Construction of the system ¶ ↺
    - depends_on lemma A.231 Restriction to an invariant closed subspace ¶ ↺
    - depends_on lemma A.234 The eigenvalues tend to zero, with finite multiplicity ¶ ↺
  - depends_on proposition 12.42 Elementary consequences ¶
    - depends_on lemma A.232 Attainment ¶ ↺
    - depends_on proposition 12.43 Norm of a self-adjoint operator ¶
      - depends_on lemma A.232 Attainment ¶ ↺
      - depends_on lemma A.239 The norm of a self-adjoint operator lies in its spectrum ¶
      - depends_on theorem A.229 Hilbert–Schmidt ¶
      - depends_on theorem A.238 Spectral theorem, both forms ¶ ↺
      - depends_on theorem 12.44 Hilbert–Schmidt: compact self-adjoint operators ¶
      - depends_on theorem 12.59 Spectral theorem for a bounded self-adjoint operator ¶ ↺
  - depends_on theorem A.229 Hilbert–Schmidt ¶ ↺
  - depends_on theorem 12.44 Hilbert–Schmidt: compact self-adjoint operators ¶ ↺
  - depends_on theorem 12.55 The spectrum of a self-adjoint operator is real ¶
    - depends_on example 12.56 Multiplication by the coordinate: spectrum without eigenvectors ¶
    - depends_on theorem A.238 Spectral theorem, both forms ¶ ↺
    - depends_on theorem 12.59 Spectral theorem for a bounded self-adjoint operator ¶ ↺
- depends_on definition 12.71 Adjoint of a densely defined operator ¶
  - depends_on definition 12.79 Deficiency subspaces and indices ¶
    - depends_on definition A.269 Cayley transform ¶
      - depends_on proposition A.270 Properties of the transform ¶
    - depends_on example 12.82 Momentum on the half-line: no self-adjoint extension ¶
      - depends_on corollary A.275 Momentum on $[0,\infty)$: no extension ¶
    - depends_on example 12.81 Momentum on a finite interval: a circle of self-adjoint momenta ¶
      - depends_on corollary A.274 Momentum on ${[}0,L{]}$: the circle of extensions ¶
      - depends_on example 12.82 Momentum on the half-line: no self-adjoint extension ¶ ↺
    - depends_on lemma A.267 Isometry of $A\pm\ii\mu$, and closed range ¶
      - depends_on definition A.269 Cayley transform ¶ ↺
      - depends_on lemma A.268 The indices do not depend on $\mu$ ¶
      - depends_on proposition A.270 Properties of the transform ¶ ↺
      - depends_on proposition A.273 Self-adjoint means unitary ¶
    - depends_on lemma A.268 The indices do not depend on $\mu$ ¶ ↺
    - depends_on theorem A.266 von Neumann ¶
      - depends_on corollary A.275 Momentum on $[0,\infty)$: no extension ¶ ↺
      - depends_on corollary A.274 Momentum on ${[}0,L{]}$: the circle of extensions ¶ ↺
    - depends_on theorem 12.80 von Neumann's criterion ¶
      - depends_on example 12.82 Momentum on the half-line: no self-adjoint extension ¶ ↺
      - depends_on example 12.81 Momentum on a finite interval: a circle of self-adjoint momenta ¶ ↺
  - depends_on definition 12.72 Symmetric; self-adjoint ¶
    - depends_on definition 12.78 Essential self-adjointness ¶
      - depends_on theorem A.266 von Neumann ¶ ↺
      - depends_on theorem 12.80 von Neumann's criterion ¶ ↺
    - depends_on lemma A.260 Cayley transform of a self-adjoint operator ¶
      - depends_on proposition A.262 Spectral theorem for an unbounded self-adjoint operator ¶
    - depends_on lemma A.267 Isometry of $A\pm\ii\mu$, and closed range ¶ ↺
    - depends_on lemma A.271 Injectivity of $\identity-V$ for any isometric extension ¶
      - depends_on lemma A.272 The operator attached to an isometry ¶
    - depends_on lemma A.272 The operator attached to an isometry ¶ ↺
    - depends_on proposition A.273 Self-adjoint means unitary ¶ ↺
    - depends_on theorem A.266 von Neumann ¶ ↺
    - depends_on theorem 12.74 Hellinger–Toeplitz ¶
      - depends_on corollary 12.76 Position and momentum are unbounded, and cannot be everywhere defined ¶
    - depends_on theorem 12.80 von Neumann's criterion ¶ ↺
  - depends_on proposition 12.73 The adjoint is always closed ¶
    - depends_on definition 12.78 Essential self-adjointness ¶ ↺
    - depends_on lemma A.260 Cayley transform of a self-adjoint operator ¶ ↺
- depends_on proposition 12.39 Algebra of the adjoint; the $C^{\ast}$ identity ¶
  - depends_on definition 12.79 Deficiency subspaces and indices ¶ ↺
  - depends_on proposition A.241 The polynomial calculus is isometric ¶
    - depends_on proposition A.243 Continuous functional calculus ¶
      - depends_on definition A.248 Cyclic vector and cyclic subspace ¶
      - depends_on proposition A.246 Bounded Borel functional calculus ¶
      - depends_on proposition A.245 The measures $\mu_{x,y}$ ¶
  - depends_on proposition 12.65 Exponential of a bounded self-adjoint operator ¶ ↺
  - depends_on proposition 12.42 Elementary consequences ¶ ↺
  - depends_on proposition 12.43 Norm of a self-adjoint operator ¶ ↺
  - depends_on proposition 12.96 Operators on a tensor product ¶
depends_on theorem 5.43 The four fundamental subspaces ¶
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 5.79 Spectral theorem for a self-adjoint operator ¶
- 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 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 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 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 definition 30.63 Flexural rigidity ¶ ↺
      - depends_on phenomenon 30.52 Euler buckling ¶
      - depends_on theorem 30.56 The beam equation ¶
    - 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 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 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 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 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 29.27 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 remark 24.42 The intermediate axis ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

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partly declared
a check failed
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declared in the source
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Edges

type	direction	node	provenance	where
`depends_on`	→	Inner product	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1875
`depends_on`	→	Linear transformation	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1875
`depends_on`	←	The adjoint exists, is unique, and is linear	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1890
`depends_on`	←	Existence and uniqueness of the adjoint	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:959
`depends_on`	←	The four fundamental subspaces	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1962
`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 self-adjoint operator	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3432