definition 5.110 Bilinear map

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:4730 · p. 151

ground object -- no derivation owed

Rests on

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 4.3 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 ¶ ↺

Supports

depends_on definition 12.94 Tensor product of Hilbert spaces ¶
- depends_on definition 12.99 Product and entangled vectors ¶
  - depends_on example 12.100 Entangled vectors exist ¶
- depends_on example 12.98 Two particles in three-dimensional space ¶
- depends_on proposition 12.95 The tensor inner product is well defined and positive definite ¶
  - depends_on example 12.100 Entangled vectors exist ¶ ↺
  - depends_on proposition 12.96 Operators on a tensor product ¶
depends_on definition 5.113 Non-degenerate form ¶
- 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 5.135 The symplectic group ¶
  - depends_on definition 22.40 Symplectic matrix ¶
  - depends_on definition 24.4 Symplectic group ¶
    - depends_on proposition 24.5 Properties of the symplectic group ¶
      - depends_on remark 24.6 Connectedness, and what it rests on ¶
      - depends_on remark 24.7 Contrast with the orthogonal group ¶
    - depends_on proposition 24.30 Linear non-squeezing ¶
      - depends_on theorem 24.31 Gromov's non-squeezing theorem ¶
  - depends_on lemma 22.20 The symplectic condition ¶
    - depends_on corollary 22.24 Invariance of the phase-space volume ¶
      - depends_on remark 22.25 This is Liouville's theorem ¶
    - depends_on proposition 22.27 Properties of the Lagrange bracket ¶
      - depends_on remark 22.28 Where the content of the fundamental brackets lies ¶
    - depends_on proposition 22.30 Properties of the Poisson bracket ¶
      - depends_on remark 22.31 The bracket is a Lie algebra structure ¶
    - depends_on remark 22.33 What the duality is good for ¶
    - depends_on remark 22.28 Where the content of the fundamental brackets lies ¶ ↺
    - depends_on theorem 22.22 Poincaré ¶
      - depends_on proposition 32.40 A Hamiltonian section preserves area ¶
      - depends_on remark 22.23 What the invariant does not depend on ¶
  - depends_on proposition 5.139 Dimension of the symplectic group ¶
    - depends_on proposition 24.5 Properties of the symplectic group ¶ ↺
    - depends_on remark 22.21 The count ¶
  - depends_on proposition 5.136 $\Sp(2n,\R)$ is a subgroup of $\GL(2n,\R)$ ¶
  - depends_on theorem 5.138 A symplectic transformation has determinant $+1$ ¶
    - depends_on corollary 22.24 Invariance of the phase-space volume ¶ ↺
    - depends_on corollary 24.19 Canonical transformations preserve phase volume ¶
    - depends_on proposition 24.5 Properties of the symplectic group ¶ ↺
- depends_on proposition 5.114 prop:lin-bilinear-dual ¶
- depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶
  - depends_on definition 5.135 The symplectic group ¶ ↺
  - depends_on definition 24.2 Symplectic vector space ¶
    - depends_on definition 24.8 Symplectic manifold ¶
      - depends_on definition 24.9 The canonical form on a cotangent bundle ¶
      - depends_on definition 24.14 Hamiltonian vector field ¶
      - depends_on definition 24.20 Liouville volume ¶
      - depends_on definition 24.53 Real polarization and polarized sections ¶
      - depends_on definition 24.49 Prequantum datum ¶
      - depends_on definition 24.17 Symplectomorphism ¶
      - depends_on lemma 24.46 The level set of the momentum map is the symplectic orthogonal of the orbit ¶
      - depends_on theorem 24.12 Darboux ¶
      - depends_on theorem 24.16 The Hamiltonian flow preserves the symplectic form ¶
      - depends_on theorem 24.38 Symplectic foliation, quoted ¶
      - depends_on theorem 24.47 Marsden–Weinstein reduction ¶
    - depends_on lemma A.564 Dimension and double orthogonal ¶
      - depends_on proposition A.565 Freeness makes every value regular ¶
      - depends_on theorem A.568 The radical is the isotropy orbit ¶
    - depends_on proposition 24.3 Even dimension and the canonical basis ¶
      - depends_on definition 24.4 Symplectic group ¶ ↺
      - depends_on proposition 26.25 The Liouville measure of a second-class surface ¶
  - depends_on proposition 26.19 Second-class constraints come in pairs ¶
    - depends_on definition 26.20 Dirac bracket ¶
      - depends_on definition 26.23 Gauge conditions ¶
      - depends_on example 26.22 Holonomic constraints ¶
      - depends_on theorem A.617 Jacobi identity for the Dirac bracket ¶
      - depends_on theorem 26.21 Properties of the Dirac bracket ¶
    - depends_on proposition 26.24 The Faddeev–Popov determinant ¶
      - depends_on remark 26.26 What the factor is doing ¶
    - depends_on proposition 26.25 The Liouville measure of a second-class surface ¶ ↺
    - depends_on theorem 26.17 Counting the physical degrees of freedom ¶
      - depends_on corollary A.642 The constraints are first class ¶
      - depends_on definition 26.23 Gauge conditions ¶ ↺
      - depends_on proposition 26.32 Constraint structure of the free Maxwell field ¶
      - depends_on proposition 26.34 Proca constraints are second class ¶
      - depends_on proposition 26.29 Reparametrization invariance and the mass-shell constraint ¶
      - depends_on theorem 26.42 The Hamiltonian structure of general relativity ¶
    - depends_on theorem 26.21 Properties of the Dirac bracket ¶ ↺
  - depends_on proposition 24.3 Even dimension and the canonical basis ¶ ↺
  - depends_on theorem A.68 Darboux ¶
    - depends_on corollary A.69 No local invariants ¶
      - depends_on remark 24.13 What Darboux's theorem forbids ¶
    - depends_on example A.80 The phase space of one particle in space ¶
    - depends_on theorem 24.12 Darboux ¶ ↺
  - depends_on theorem 5.138 A symplectic transformation has determinant $+1$ ¶ ↺
depends_on definition 5.111 Symmetric and antisymmetric forms ¶
- depends_on definition 28.32 The two quadratic forms ¶
- depends_on proposition 5.112 Symmetric–antisymmetric splitting ¶
- depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶ ↺
- depends_on theorem 5.116 Sylvester's law of inertia ¶
  - depends_on definition 5.117 Signature ¶ ↺
  - 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 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 proposition 10.13 Canonical form in two variables ¶
      - depends_on proposition 10.11 What a characteristic surface is ¶
  - 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 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 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 ¶ ↺
depends_on lemma 5.137 The alternating top form is unique up to scale ¶
- depends_on theorem 5.138 A symplectic transformation has determinant $+1$ ¶ ↺
depends_on proposition 5.114 prop:lin-bilinear-dual ¶ ↺
depends_on proposition 5.112 Symmetric–antisymmetric splitting ¶ ↺
depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶ ↺
depends_on theorem 5.116 Sylvester's law of inertia ¶ ↺

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	Linear transformation	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4744
`depends_on`	←	Tensor product of Hilbert spaces	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2596
`depends_on`	←	Non-degenerate form	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4880
`depends_on`	←	Symmetric and antisymmetric forms	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4827
`depends_on`	←	The alternating top form is unique up to scale	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:5985
`depends_on`	←	prop:lin-bilinear-dual	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4899
`depends_on`	←	Symmetric–antisymmetric splitting	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4841
`depends_on`	←	Normal form of a non-degenerate antisymmetric form	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:5106
`depends_on`	←	Sylvester's law of inertia	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4998