lemma A.601 Several variables, complex symmetric matrix

open in the book · appendices/A-long-proofs.tex:28760 · p. 3080

Rests on

depends_on lemma A.599 One variable, complex coefficient ¶
- depends_on equation 17.42 eq:ft-gaussian ¶
- depends_on theorem 7.109 Leibniz integral rule ¶
  - depends_on definition 7.125 Multiple integral ¶
    - depends_on axiom 7.1 Completeness of $\R$ ¶
    - depends_on definition 7.39 Darboux sums and the definite integral ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
  - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
    - depends_on proposition 7.22 Sequential characterization ¶
      - depends_on definition 7.20 Continuity at a point ¶
      - depends_on definition 7.4 Convergence ¶
      - proves proof ch:05-real-analysis@proof-7 ¶
    - depends_on theorem 7.7 Bolzano–Weierstrass ¶
      - depends_on corollary A.45 Archimedean property and density of $\Q$ ¶
      - depends_on corollary A.46 Monotone convergence ¶
      - proves proof ch:05-real-analysis@proof-4 ¶
    - proves proof ch:05-real-analysis@proof-10 ¶
  - depends_on theorem 7.35 Mean value theorem ¶
    - depends_on proposition 7.29 Linearity ¶
      - depends_on definition 7.11 Sum of functions ¶
      - depends_on equation 7.14 eq:ana-derivh ¶
      - depends_on proposition 7.6 Algebra of limits ¶
      - proves proof ch:05-real-analysis@proof-13 ¶
    - depends_on theorem 7.34 Rolle ¶
      - depends_on lemma 7.33 Fermat: interior extremum ¶
      - depends_on theorem 7.24 Extreme value theorem ¶
      - proves proof ch:05-real-analysis@proof-18 ¶
    - proves proof ch:05-real-analysis@proof-19 ¶
  - proves proof ch:05-real-analysis@proof-67 ¶
- depends_on theorem 8.6 Cauchy–Riemann equations ¶
  - depends_on definition 7.98 Functions of class $C^{1}$ ¶
    - depends_on definition 7.20 Continuity at a point ¶ ↺
    - depends_on definition 7.97 Partial derivative; gradient ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on definition 5.15 Basis ¶
  - depends_on equation 8.4 eq:cpx-derivative ¶
  - proves proof ch:06-complex-analysis@proof-3 ¶
- proves proof app:A-long-proofs@proof-358 ¶
depends_on lemma A.600 The real part of an inverse ¶
- depends_on theorem 5.40 Rank–nullity ¶
  - depends_on definition 5.15 Basis ¶ ↺
  - depends_on definition 5.39 Kernel, image, nullity, rank ¶
    - depends_on definition 5.37 Linear transformation ¶
      - 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-12 ¶
  - 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 5.5 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 5.5 Linear combination ¶ ↺
    - proves proof ch:03-linear-algebra-representations@proof-11 ¶
  - proves proof ch:03-linear-algebra-representations@proof-13 ¶
- depends_on theorem 5.84 Spectral theorem for a real symmetric operator ¶
  - depends_on proposition 5.28 Gram–Schmidt ¶
    - depends_on definition 5.14 Linear independence ¶ ↺
    - 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 5.25 Unit vector ¶
    - 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 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 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 theorem 5.40 Rank–nullity ¶ ↺
      - 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 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 ¶
- proves proof app:A-long-proofs@proof-359 ¶
depends_on theorem 5.84 Spectral theorem for a real symmetric operator ¶ ↺
proves proof app:A-long-proofs@proof-360 ¶

Supports

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 proposition A.612 The dictionary between $A$ and $\Gamma$ ¶ ↺

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	One variable, complex coefficient	declared	appendices/A-long-proofs.tex:28772
`depends_on`	→	The real part of an inverse	declared	appendices/A-long-proofs.tex:28772
`depends_on`	→	Spectral theorem for a real symmetric operator	declared	appendices/A-long-proofs.tex:28772
`depends_on`	←	The dictionary between $A$ and $\Gamma$	declared	appendices/A-long-proofs.tex:29361
`depends_on`	←	The Wigner function of a Gaussian is a positive Gaussian	declared	appendices/A-long-proofs.tex:28831
`proves`	←	app:A-long-proofs@proof-360	declared	appendices/A-long-proofs.tex:28776