definition 5.26 Orthogonal basis

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:1258 · p. 111

ground object -- no derivation owed

Rests on

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 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.32 Field ¶
      - depends_on definition 4.31 Module ¶
  - 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 ¶ ↺

Supports

depends_on definition 5.27 Orthonormal basis ¶
- 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 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 theorem A.229 Hilbert–Schmidt ¶
      - depends_on theorem 12.44 Hilbert–Schmidt: compact self-adjoint operators ¶
      - 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 lemma 5.132 Parity constraint ¶
      - depends_on theorem 5.133 Isotropic Cartesian tensors of rank at most four ¶
  - 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 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 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 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 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.

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`	→	Basis	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1265
`depends_on`	→	Orthogonal vectors	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1265
`depends_on`	←	Orthonormal basis	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1276