lemma 5.150 Invariance of the orthogonal complement

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:6374 · p. 171

Rests on

depends_on definition 5.147 Invariant subspace ¶
- depends_on definition 5.7 Vector subspace ¶
  - depends_on definition 5.5 Linear combination ¶
    - depends_on definition 4.33 Vector space ¶
      - depends_on definition 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.32 Field ¶
      - depends_on definition 4.31 Module ¶
- depends_on definition 5.141 Representation of a group ¶
  - depends_on definition 4.21 Group ¶
    - 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 4.43 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 4.33 Vector space ¶ ↺
- depends_on definition 5.141 Representation of a group ¶ ↺
- depends_on equation 5.171 eq:rep-unitary-group ¶
depends_on equation 4.25 eq:alg-hom-inverse ¶
depends_on equation 5.52 eq:lin-orthogonal-complement ¶
depends_on equation 5.183 eq:rep-unitary-invariance ¶
proves proof ch:03-linear-algebra-representations@proof-63 ¶

Supports

depends_on proposition 5.151 prop:rep-not-totally-reducible ¶
depends_on proposition 5.153 prop:rep-unitary-completely-reducible ¶
- 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`	→	Invariant subspace	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6382
`depends_on`	→	Unitary representation	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6382
`depends_on`	→	eq:alg-hom-inverse	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6382
`depends_on`	→	eq:lin-orthogonal-complement	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6382
`depends_on`	→	eq:rep-unitary-invariance	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6382
`depends_on`	←	prop:rep-not-totally-reducible	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6413
`depends_on`	←	prop:rep-unitary-completely-reducible	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6485
`proves`	←	ch:03-linear-algebra-representations@proof-63	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6385