proposition 5.94 Principal axes of a real quadratic form

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:4094 · p. 144

Rests on

depends_on definition 5.117 Signature ¶
- depends_on definition 5.113 Non-degenerate form ¶
  - depends_on definition 5.110 Bilinear map ¶
    - depends_on definition 5.37 Linear transformation ¶
      - depends_on definition 4.33 Vector space ¶
- depends_on theorem 5.116 Sylvester's law of inertia ¶
  - depends_on definition 5.110 Bilinear map ¶ ↺
  - depends_on definition 5.111 Symmetric and antisymmetric forms ¶
    - depends_on definition 5.110 Bilinear map ¶ ↺
    - depends_on equation 5.137 eq:lin-bilinear-matrix ¶
  - depends_on equation 5.139 eq:lin-bilinear-congruence ¶
  - proves proof ch:03-linear-algebra-representations@proof-53 ¶
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 5.5 Linear combination ¶
      - depends_on definition 4.33 Vector space ¶ ↺
  - 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 definition 5.37 Linear transformation ¶ ↺
      - depends_on definition 5.7 Vector subspace ¶
      - 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 ¶ ↺
      - 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 ¶
      - 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 ¶
      - 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 ¶
depends_on theorem 5.116 Sylvester's law of inertia ¶ ↺
proves proof ch:03-linear-algebra-representations@proof-41 ¶

Supports

depends_on proposition 5.95 The focal polar equation of a conic ¶

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`	→	Signature	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4107
`depends_on`	→	Spectral theorem for a real symmetric operator	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4107
`depends_on`	→	Sylvester's law of inertia	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4107
`depends_on`	←	The focal polar equation of a conic	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4153
`proves`	←	ch:03-linear-algebra-representations@proof-41	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4110