theorem 5.86 Simultaneous diagonalization of a definite pencil

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:3761 · p. 141

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 5.84 Spectral theorem for a real symmetric operator ¶
- depends_on proposition 5.28 Gram–Schmidt ¶ ↺
- 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 ¶
- depends_on definition 5.110 Bilinear map ¶
  - depends_on definition 5.37 Linear transformation ¶ ↺
- 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 ¶
proves proof ch:03-linear-algebra-representations@proof-35 ¶

Supports

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 proposition 5.92 The eigenvalues do not control the norm ¶
- depends_on corollary 5.89 Rayleigh–Ritz ¶ ↺

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	Gram–Schmidt	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3781
`depends_on`	→	Spectral theorem for a real symmetric operator	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3781
`depends_on`	→	Sylvester's law of inertia	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3781
`depends_on`	←	Rayleigh–Ritz	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3929
`depends_on`	←	Rayleigh quotient of a definite pencil	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3886
`proves`	←	ch:03-linear-algebra-representations@proof-35	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3785