proposition 5.155 Averaging trick

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:6581 · p. 173

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 5.145 Equivalent representations ¶
- depends_on definition 5.141 Representation of a group ¶
  - depends_on definition 4.21 Group ¶ ↺
  - depends_on definition 4.43 Group homomorphism ¶
    - depends_on definition 4.21 Group ¶ ↺
    - depends_on definition 3.43 Map ¶
      - depends_on definition 3.24 Quantifiers ¶
      - depends_on definition 3.28 Set ¶
  - depends_on definition 4.48 Symmetric group ¶
    - depends_on definition 3.47 Bijective map ¶
depends_on definition 5.144 Unitary representation ¶
- depends_on definition 5.18 Inner product ¶ ↺
- depends_on definition 5.141 Representation of a group ¶ ↺
- depends_on equation 5.171 eq:rep-unitary-group ¶
depends_on equation 5.183 eq:rep-unitary-invariance ¶
depends_on proposition 5.28 Gram–Schmidt ¶
- depends_on definition 5.14 Linear independence ¶
  - depends_on definition 5.5 Linear combination ¶
    - depends_on definition 4.33 Vector space ¶ ↺
- depends_on definition 5.18 Inner product ¶ ↺
- depends_on definition 5.27 Orthonormal basis ¶
  - depends_on definition 5.26 Orthogonal basis ¶
    - depends_on definition 5.15 Basis ¶
      - depends_on definition 5.12 Subspace generated by a set of vectors ¶
      - depends_on definition 5.14 Linear independence ¶ ↺
    - depends_on definition 5.24 Orthogonal vectors ¶
      - depends_on definition 5.18 Inner product ¶ ↺
  - depends_on definition 5.25 Unit vector ¶
    - depends_on definition 5.19 Norm ¶
      - depends_on definition 4.33 Vector space ¶ ↺
- depends_on equation 5.44 eq:lin-norm-assoc ¶
- proves proof ch:03-linear-algebra-representations@proof-5 ¶
proves proof ch:03-linear-algebra-representations@proof-66 ¶

Supports

depends_on theorem 101.62 The mixing matrix is unitary ¶
- depends_on proposition 101.63 Parameter counting, and why three generations permit $CP$ violation ¶
- depends_on theorem 103.11 GIM suppression at one loop ¶
  - depends_on phenomenon 103.42 $CP$ violation in charm decays ¶

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`	→	Inner product	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6603
`depends_on`	→	Equivalent representations	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6603
`depends_on`	→	Unitary representation	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6603
`depends_on`	→	eq:rep-unitary-invariance	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6603
`depends_on`	→	Gram–Schmidt	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6603
`depends_on`	←	The mixing matrix is unitary	declared	parts/11-qft-standard-model/03-weak-interactions.tex:3000
`proves`	←	ch:03-linear-algebra-representations@proof-66	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6606