theorem 5.84 Spectral theorem for a real symmetric operator

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:3683 · p. 140

Rests on

depends_on proposition 5.28 Gram–Schmidt ¶
- 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 4.32 Field ¶
      - depends_on definition 4.31 Module ¶
- 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 5.14 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 ¶
depends_on theorem 5.71 The eigenvalues are the roots of the characteristic polynomial ¶
- depends_on definition 5.70 Characteristic polynomial ¶
  - depends_on definition 5.56 Endomorphism ¶
    - depends_on definition 5.37 Linear transformation ¶
      - depends_on definition 4.33 Vector space ¶ ↺
  - depends_on equation 5.19 eq:lin-leibniz-det ¶
  - depends_on equation 5.74 eq:lin-matrix-rep ¶
  - proves proof ch:03-linear-algebra-representations@proof-25 ¶
- depends_on definition 5.69 Eigenvector, eigenvalue, eigenspace ¶
  - depends_on definition 5.56 Endomorphism ¶ ↺
  - depends_on definition 5.39 Kernel, image, nullity, rank ¶
    - depends_on definition 5.37 Linear transformation ¶ ↺
    - depends_on definition 5.7 Vector subspace ¶
      - depends_on definition 5.5 Linear combination ¶ ↺
    - proves proof ch:03-linear-algebra-representations@proof-12 ¶
- depends_on proposition 5.49 Injective, surjective, invertible ¶
  - depends_on definition 5.39 Kernel, image, nullity, rank ¶ ↺
  - depends_on proposition 5.48 Existence, uniqueness, and linearity of the inverse ¶
    - depends_on definition 5.47 Inverse of a linear transformation ¶
      - depends_on definition 5.37 Linear transformation ¶ ↺
    - depends_on definition 5.37 Linear transformation ¶ ↺
    - proves proof ch:03-linear-algebra-representations@proof-17 ¶
  - depends_on theorem 5.40 Rank–nullity ¶
    - depends_on definition 5.15 Basis ¶ ↺
    - depends_on definition 5.39 Kernel, image, nullity, rank ¶ ↺
    - depends_on lemma 5.38 Exchange and completion ¶
      - depends_on definition 5.15 Basis ¶ ↺
      - depends_on definition 5.12 Subspace generated by a set of vectors ¶ ↺
      - depends_on definition 5.14 Linear independence ¶ ↺
      - proves proof ch:03-linear-algebra-representations@proof-11 ¶
    - proves proof ch:03-linear-algebra-representations@proof-13 ¶
  - proves proof ch:03-linear-algebra-representations@proof-18 ¶
- proves proof ch:03-linear-algebra-representations@proof-26 ¶
depends_on theorem 5.79 Spectral theorem for a self-adjoint operator ¶
- depends_on corollary 5.72 Existence of an eigenvalue over $\C$ ¶
  - depends_on theorem 8.19 Fundamental theorem of algebra ¶
    - depends_on theorem 8.18 Liouville ¶
      - depends_on proposition 8.10 Fundamental theorem for contours ¶
      - depends_on theorem 8.17 Derivatives of all orders; Cauchy estimates ¶
      - proves proof ch:06-complex-analysis@proof-12 ¶
    - proves proof ch:06-complex-analysis@proof-13 ¶
  - depends_on theorem 5.71 The eigenvalues are the roots of the characteristic polynomial ¶ ↺
  - proves proof ch:03-linear-algebra-representations@proof-27 ¶
- depends_on definition 5.41 Adjoint ¶
  - depends_on definition 5.18 Inner product ¶ ↺
  - depends_on definition 5.37 Linear transformation ¶ ↺
- depends_on theorem 5.40 Rank–nullity ¶ ↺
- proves proof ch:03-linear-algebra-representations@proof-31 ¶
proves proof ch:03-linear-algebra-representations@proof-34 ¶

Supports

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 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 proposition A.611 The constraint on $\Gamma$ ¶ ↺
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 ¶
- 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 ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
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inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Gram–Schmidt	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3697
`depends_on`	→	The eigenvalues are the roots of the characteristic polynomial	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3697
`depends_on`	→	Spectral theorem for a self-adjoint operator	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3697
`depends_on`	←	The operator norm of a real array	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4006
`depends_on`	←	Block positivity	declared	appendices/A-long-proofs.tex:29245
`depends_on`	←	Several variables, complex symmetric matrix	declared	appendices/A-long-proofs.tex:28772
`depends_on`	←	The real part of an inverse	declared	appendices/A-long-proofs.tex:28744
`depends_on`	←	Polar decomposition	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3958
`depends_on`	←	Principal axes of a real quadratic form	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4107
`depends_on`	←	The eigenvalues do not control the norm	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4050
`depends_on`	←	Simultaneous diagonalization of a definite pencil	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3781
`proves`	←	ch:03-linear-algebra-representations@proof-34	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3701