theorem 5.98 Primary decomposition

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:4292 · p. 147

Rests on

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 equation 8.6 eq:cpx-contour-integral ¶
      - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
      - proves proof ch:06-complex-analysis@proof-5 ¶
    - depends_on theorem 8.17 Derivatives of all orders; Cauchy estimates ¶
      - depends_on lemma 8.9 ML estimate ¶
      - depends_on theorem 8.16 Cauchy integral formula ¶
      - proves proof ch:06-complex-analysis@proof-11 ¶
    - 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 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 ¶
- proves proof ch:03-linear-algebra-representations@proof-27 ¶
depends_on equation 5.102 eq:lin-eigenspace ¶
depends_on lemma 5.97 Fitting splitting ¶
- depends_on definition 5.39 Kernel, image, nullity, rank ¶ ↺
- depends_on proposition 5.49 Injective, surjective, invertible ¶ ↺
- depends_on theorem 5.40 Rank–nullity ¶ ↺
- proves proof ch:03-linear-algebra-representations@proof-43 ¶
proves proof ch:03-linear-algebra-representations@proof-44 ¶

Supports

depends_on corollary 5.99 Semisimple and nilpotent parts ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
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declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Existence of an eigenvalue over $\C$	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4308
`depends_on`	→	eq:lin-eigenspace	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4308
`depends_on`	→	Fitting splitting	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4308
`depends_on`	←	Semisimple and nilpotent parts	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4372
`proves`	←	ch:03-linear-algebra-representations@proof-44	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4312