theorem 5.80 Simultaneous diagonalization of commuting self-adjoint operators

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:3504 · p. 138

Rests on

depends_on definition 5.41 Adjoint ¶
- depends_on definition 5.18 Inner product ¶
  - depends_on definition 4.33 Vector space ¶
    - depends_on definition 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.4 Internal binary operation; magma ¶
    - depends_on definition 4.32 Field ¶
      - depends_on definition 4.11 Inverse element ¶
      - depends_on definition 4.20 Monoid ¶
      - depends_on definition 4.30 Ring ¶
    - depends_on definition 4.31 Module ¶
      - depends_on definition 4.29 Action ¶
      - depends_on definition 4.28 Distributivity ¶
      - depends_on definition 4.21 Group ¶
      - depends_on definition 4.30 Ring ¶ ↺
- depends_on definition 5.37 Linear transformation ¶
  - depends_on definition 4.33 Vector space ¶ ↺
depends_on equation 5.102 eq:lin-eigenspace ¶
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 ¶
      - depends_on proposition 8.10 Fundamental theorem for contours ¶
      - depends_on theorem 8.17 Derivatives of all orders; Cauchy estimates ¶
      - proves proof ch:06-complex-analysis@proof-12 ¶
    - proves proof ch:06-complex-analysis@proof-13 ¶
  - 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 ¶
  - proves proof ch:03-linear-algebra-representations@proof-27 ¶
- 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-32 ¶

Supports

depends_on corollary 5.82 Spectral theorem for a normal operator ¶

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`	→	Adjoint	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3513
`depends_on`	→	eq:lin-eigenspace	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3513
`depends_on`	→	Spectral theorem for a self-adjoint operator	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3513
`depends_on`	←	Spectral theorem for a normal operator	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3594
`proves`	←	ch:03-linear-algebra-representations@proof-32	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3516