proposition 5.28 Gram–Schmidt

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:1301 · p. 112

Rests on

depends_on definition 5.14 Linear independence ¶
- 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 4.4 Internal binary operation; magma ¶
    - depends_on definition 4.32 Field ¶
      - depends_on definition 4.11 Inverse element ¶
      - depends_on definition 4.20 Monoid ¶
      - depends_on definition 4.30 Ring ¶
    - depends_on definition 4.31 Module ¶
      - depends_on definition 4.29 Action ¶
      - depends_on definition 4.28 Distributivity ¶
      - depends_on definition 4.21 Group ¶
      - depends_on definition 4.30 Ring ¶ ↺
depends_on definition 5.18 Inner product ¶
- depends_on definition 4.33 Vector space ¶ ↺
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 5.5 Linear combination ¶ ↺
      - depends_on definition 5.7 Vector subspace ¶
      - proves proof ch:03-linear-algebra-representations@proof-3 ¶
    - 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 ¶

Supports

depends_on corollary 5.29 Orthogonal decomposition ¶
- 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 ¶
- depends_on theorem 5.43 The four fundamental subspaces ¶
depends_on proposition 12.23 Gram–Schmidt in a Hilbert space ¶
- depends_on corollary 12.24 Separable spaces have countable orthonormal families ¶
  - depends_on theorem 12.33 Every separable Hilbert space is $\ell^{2}$ ¶
- depends_on example 12.25 Orthogonal polynomials ¶
- 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 proposition 5.42 The adjoint exists, is unique, and is linear ¶
- depends_on corollary 5.82 Spectral theorem for a normal operator ¶
depends_on proposition 5.155 Averaging trick ¶
- depends_on theorem 101.62 The mixing matrix is unitary ¶ ↺
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 proposition 5.92 The eigenvalues do not control the norm ¶
  - depends_on corollary 5.89 Rayleigh–Ritz ¶ ↺
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 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 proposition A.609 The exponent is a quadratic polynomial ¶
- 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 ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

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partly declared
a check failed
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Edges

type	direction	node	provenance	where
`depends_on`	→	Linear independence	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1312
`depends_on`	→	Inner product	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1312
`depends_on`	→	Orthonormal basis	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1312
`depends_on`	→	eq:lin-norm-assoc	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1312
`depends_on`	←	Orthogonal decomposition	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1363
`depends_on`	←	Gram–Schmidt in a Hilbert space	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:587
`depends_on`	←	The adjoint exists, is unique, and is linear	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1890
`depends_on`	←	Averaging trick	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6603
`depends_on`	←	Simultaneous diagonalization of a definite pencil	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3781
`depends_on`	←	Spectral theorem for a real symmetric operator	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3697
`proves`	←	ch:03-linear-algebra-representations@proof-5	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1315