theorem 5.138 A symplectic transformation has determinant $+1$

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:6032 · p. 167

Rests on

depends_on definition 5.135 The symplectic group ¶
- depends_on definition 5.113 Non-degenerate form ¶
  - depends_on definition 5.110 Bilinear map ¶
    - depends_on definition 5.37 Linear transformation ¶
      - depends_on definition 4.33 Vector space ¶
- depends_on equation 5.138 eq:lin-bilinear-expansion ¶
- depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶
  - depends_on definition 5.110 Bilinear map ¶ ↺
  - depends_on definition 5.113 Non-degenerate form ¶ ↺
  - 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-54 ¶
depends_on lemma 5.137 The alternating top form is unique up to scale ¶
- depends_on definition 5.15 Basis ¶
  - depends_on definition 5.12 Subspace generated by a set of vectors ¶
    - depends_on definition 5.5 Linear combination ¶
      - depends_on definition 4.33 Vector space ¶ ↺
    - depends_on definition 5.7 Vector subspace ¶
      - depends_on definition 5.5 Linear combination ¶ ↺
    - proves proof ch:03-linear-algebra-representations@proof-3 ¶
  - depends_on definition 5.14 Linear independence ¶
    - depends_on definition 5.5 Linear combination ¶ ↺
- depends_on definition 5.110 Bilinear map ¶ ↺
- depends_on equation 5.19 eq:lin-leibniz-det ¶
- proves proof ch:03-linear-algebra-representations@proof-60 ¶
depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶ ↺
proves proof ch:03-linear-algebra-representations@proof-61 ¶

Supports

depends_on corollary 22.24 Invariance of the phase-space volume ¶
- depends_on remark 22.25 This is Liouville's theorem ¶
depends_on corollary 24.19 Canonical transformations preserve phase volume ¶
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 ¶

Neighborhood

Every logical edge within two steps of this node.

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partly declared
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declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	The symplectic group	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6036
`depends_on`	→	The alternating top form is unique up to scale	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6036
`depends_on`	→	Normal form of a non-degenerate antisymmetric form	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6036
`depends_on`	←	Invariance of the phase-space volume	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:800
`depends_on`	←	Canonical transformations preserve phase volume	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:622
`depends_on`	←	Properties of the symplectic group	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:165
`proves`	←	ch:03-linear-algebra-representations@proof-61	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6040