definition 5.27 Orthonormal basis

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:1268 · p. 112

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Rests on

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 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 5.24 Orthogonal vectors ¶
  - depends_on definition 5.18 Inner product ¶
    - depends_on definition 4.33 Vector space ¶ ↺
depends_on definition 5.25 Unit vector ¶
- depends_on definition 5.19 Norm ¶
  - depends_on definition 4.33 Vector space ¶ ↺

Supports

depends_on definition 12.26 Orthonormal system; Fourier coefficients ¶
- depends_on definition 12.29 Orthonormal basis ¶
  - depends_on definition 12.86 Internal orthogonal decomposition ¶
    - depends_on proposition 12.87 Expansion in an orthogonal decomposition ¶
      - depends_on lemma A.250 Decomposition into cyclic subspaces ¶
- depends_on proposition 12.27 Best approximation and Bessel's inequality ¶
  - depends_on theorem 12.30 Completeness, expansion, Parseval ¶
    - depends_on proposition 17.26 The lattice harmonics are an orthonormal basis ¶
    - depends_on proposition 12.87 Expansion in an orthogonal decomposition ¶ ↺
    - 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 theorem A.229 Hilbert–Schmidt ¶
    - depends_on theorem 12.44 Hilbert–Schmidt: compact self-adjoint operators ¶
      - depends_on theorem A.461 Completeness in the weighted and in the energy norm ¶
      - depends_on theorem A.471 Spectral decomposition and completeness in $L^{2}_{r}$ ¶
    - depends_on theorem 12.33 Every separable Hilbert space is $\ell^{2}$ ¶
- depends_on proposition 12.28 Convergence criterion for orthogonal series ¶
  - depends_on theorem 12.30 Completeness, expansion, Parseval ¶ ↺
depends_on definition 5.32 Levi–Civita symbol; cross product ¶
- depends_on definition 5.131 Isotropic Cartesian tensor ¶
  - depends_on definition A.672 The octahedral rotation group ¶
    - depends_on proposition A.673 Cubic tensors of rank four ¶
      - depends_on theorem A.675 Three constants for a cubic crystal, two for an isotropic solid ¶
  - depends_on lemma 5.132 Parity constraint ¶
    - depends_on theorem 5.133 Isotropic Cartesian tensors of rank at most four ¶
      - depends_on lemma A.742 Isotropic representation ¶
      - depends_on lemma A.671 The three products are independent ¶
      - depends_on lemma 30.26 Isotropic Cartesian tensors of rank four ¶
      - depends_on remark A.677 The lower ranks, and why an isotropic solid is not piezoelectric ¶
      - depends_on remark 30.27 Two results this chapter borrows from Part II ¶
      - depends_on remark 13.7 What the classification is used for ¶
- depends_on lemma 5.34 Contraction of two Levi–Civita symbols ¶
  - depends_on proposition 5.35 The identities of the vector algebra of $\R^{3}$ ¶
- depends_on lemma 5.33 Determinant through the Levi–Civita symbol ¶
  - depends_on proposition 5.35 The identities of the vector algebra of $\R^{3}$ ¶ ↺
  - depends_on theorem 5.133 Isotropic Cartesian tensors of rank at most four ¶ ↺
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 proposition 5.28 Gram–Schmidt ¶
- 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 theorem 5.43 The four fundamental subspaces ¶
- depends_on proposition 12.23 Gram–Schmidt in a Hilbert space ¶ ↺
- 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 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

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Edges

type	direction	node	provenance	where
`depends_on`	→	Orthogonal basis	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1276
`depends_on`	→	Unit vector	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1276
`depends_on`	←	Orthonormal system; Fourier coefficients	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:657
`depends_on`	←	Levi–Civita symbol; cross product	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1489
`depends_on`	←	Gram–Schmidt in a Hilbert space	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:587
`depends_on`	←	Gram–Schmidt	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1312