definition 5.24 Orthogonal vectors

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:1241 · p. 111

ground object -- no derivation owed

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 ¶ ↺

Supports

depends_on corollary 12.5 Continuity of the norm and of orthogonality ¶
- depends_on proposition 12.17 The complement is always a closed subspace ¶
  - depends_on corollary 12.19 Double complement; the density criterion ¶
    - depends_on definition 12.79 Deficiency subspaces and indices ¶
      - depends_on definition A.269 Cayley transform ¶
      - depends_on example 12.82 Momentum on the half-line: no self-adjoint extension ¶
      - depends_on example 12.81 Momentum on a finite interval: a circle of self-adjoint momenta ¶
      - depends_on lemma A.267 Isometry of $A\pm\ii\mu$, and closed range ¶
      - depends_on lemma A.268 The indices do not depend on $\mu$ ¶
      - depends_on theorem A.266 von Neumann ¶
      - depends_on theorem 12.80 von Neumann's criterion ¶
    - depends_on definition 12.69 Operator with a domain ¶
      - depends_on definition 12.70 Graph; closed and closable operators ¶
      - depends_on definition 12.72 Symmetric; self-adjoint ¶
      - depends_on definition 12.71 Adjoint of a densely defined operator ¶
    - depends_on definition 12.103 Gelfand triple ¶
      - depends_on definition A.278 Countably Hilbert nuclear space ¶
      - depends_on definition 12.105 Generalized eigenvector ¶
      - depends_on example 12.104 The Schwartz triple ¶
      - depends_on theorem A.279 Gelfand–Maurin ¶
      - depends_on theorem 12.107 Nuclear spectral theorem ¶
    - depends_on definition 12.50 Point, continuous and residual spectrum ¶
      - depends_on definition 12.105 Generalized eigenvector ¶ ↺
      - depends_on example 12.56 Multiplication by the coordinate: spectrum without eigenvectors ¶
      - depends_on proposition 12.51 The three cases are exclusive and exhaustive ¶
    - depends_on definition 12.71 Adjoint of a densely defined operator ¶ ↺
    - 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 theorem 12.55 The spectrum of a self-adjoint operator is real ¶
      - depends_on example 12.56 Multiplication by the coordinate: spectrum without eigenvectors ¶ ↺
      - depends_on theorem A.238 Spectral theorem, both forms ¶
      - depends_on theorem 12.59 Spectral theorem for a bounded self-adjoint operator ¶
  - depends_on theorem 12.18 Projection theorem ¶
    - depends_on corollary 12.19 Double complement; the density criterion ¶ ↺
    - depends_on definition A.248 Cyclic vector and cyclic subspace ¶
      - depends_on lemma A.249 The cyclic case ¶
      - depends_on lemma A.250 Decomposition into cyclic subspaces ¶
    - depends_on definition 12.20 Orthogonal projection operator ¶
      - depends_on definition 12.88 Reducing subspace ¶
      - depends_on proposition 12.21 Characterization of orthogonal projections ¶
    - depends_on lemma A.233 Construction of the system ¶
      - depends_on lemma A.234 The eigenvalues tend to zero, with finite multiplicity ¶
    - depends_on lemma A.231 Restriction to an invariant closed subspace ¶
      - depends_on lemma A.233 Construction of the system ¶ ↺
    - depends_on lemma A.267 Isometry of $A\pm\ii\mu$, and closed range ¶ ↺
    - depends_on proposition 12.27 Best approximation and Bessel's inequality ¶
      - depends_on theorem 12.30 Completeness, expansion, Parseval ¶ ↺
    - depends_on proposition 12.21 Characterization of orthogonal projections ¶ ↺
    - depends_on theorem 12.46 Riesz representation ¶
      - depends_on corollary 12.47 $\mathcal{H}$ is its own dual, antilinearly ¶
      - depends_on lemma A.247 Integration against a projection-valued measure ¶
      - depends_on proposition A.246 Bounded Borel functional calculus ¶
      - depends_on proposition A.245 The measures $\mu_{x,y}$ ¶
      - depends_on theorem 12.38 Existence and uniqueness of the adjoint ¶
depends_on definition 12.16 Orthogonal complement ¶
- 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 definition 12.86 Internal orthogonal decomposition ¶ ↺
- depends_on proposition 12.17 The complement is always a closed subspace ¶ ↺
depends_on definition 5.26 Orthogonal basis ¶
- 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 proposition 12.27 Best approximation and Bessel's inequality ¶ ↺
    - 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 lemma 5.132 Parity constraint ¶
    - 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 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 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 lemma A.601 Several variables, complex symmetric matrix ¶
      - depends_on lemma A.600 The real part of an inverse ¶
      - depends_on proposition 5.90 Polar decomposition ¶
      - depends_on proposition 5.94 Principal axes of a real quadratic form ¶
      - 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
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Edges

type	direction	node	provenance	where
`depends_on`	→	Inner product	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1248
`depends_on`	←	Continuity of the norm and of orthogonality	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:146
`depends_on`	←	Orthogonal complement	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:398
`depends_on`	←	Orthogonal basis	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1265