proposition A.602 The Wigner function of a Gaussian is a positive Gaussian

open in the book · appendices/A-long-proofs.tex:28815 · p. 3080

Rests on

depends_on equation A.910 eq:app-hudson-gaussian-form ¶
depends_on equation 25.53 eq:pq-wigner ¶
depends_on lemma A.601 Several variables, complex symmetric matrix ¶
- 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 theorem 7.25 Heine–Cantor: uniform continuity ¶
      - depends_on proposition 7.22 Sequential characterization ¶
      - depends_on theorem 7.7 Bolzano–Weierstrass ¶
      - proves proof ch:05-real-analysis@proof-10 ¶
    - depends_on theorem 7.35 Mean value theorem ¶
      - depends_on proposition 7.29 Linearity ¶
      - depends_on theorem 7.34 Rolle ¶
      - 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 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 5.12 Subspace generated by a set of vectors ¶
      - depends_on definition 5.14 Linear independence ¶
    - depends_on definition 5.39 Kernel, image, nullity, rank ¶
      - depends_on definition 5.37 Linear transformation ¶
      - depends_on definition 5.7 Vector subspace ¶
      - 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 5.14 Linear independence ¶ ↺
      - 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 5.27 Orthonormal basis ¶
      - 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 5.69 Eigenvector, eigenvalue, eigenspace ¶
      - depends_on proposition 5.49 Injective, surjective, invertible ¶
      - 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 definition 5.41 Adjoint ¶
      - 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 ¶
proves proof app:A-long-proofs@proof-361 ¶

Supports

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

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`	→	eq:app-hudson-gaussian-form	declared	appendices/A-long-proofs.tex:28831
`depends_on`	→	eq:pq-wigner	declared	appendices/A-long-proofs.tex:28831
`depends_on`	→	Several variables, complex symmetric matrix	declared	appendices/A-long-proofs.tex:28831
`depends_on`	←	Non-negativity forbids a vanishing overlap	declared	appendices/A-long-proofs.tex:28961
`proves`	←	app:A-long-proofs@proof-361	declared	appendices/A-long-proofs.tex:28835