lemma A.610 Block positivity

open in the book · appendices/A-long-proofs.tex:29235 · p. 3084

Rests on

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 5.5 Linear combination ¶
      - 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-3 ¶
  - depends_on definition 5.14 Linear independence ¶
    - depends_on definition 5.5 Linear combination ¶ ↺
- 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 ¶ ↺
  - 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 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 5.24 Orthogonal vectors ¶
    - depends_on definition 5.25 Unit vector ¶
      - depends_on definition 5.19 Norm ¶
  - 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 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 definition 5.47 Inverse of a 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 ¶ ↺
    - 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 ¶
      - 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 ¶
proves proof app:A-long-proofs@proof-368 ¶

Supports

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`	→	Rank–nullity	declared	appendices/A-long-proofs.tex:29245
`depends_on`	→	Spectral theorem for a real symmetric operator	declared	appendices/A-long-proofs.tex:29245
`depends_on`	←	The constraint on $\Gamma$	declared	appendices/A-long-proofs.tex:29285
`proves`	←	app:A-long-proofs@proof-368	declared	appendices/A-long-proofs.tex:29248