definition 5.56 Endomorphism

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:2491 · p. 126

ground object -- no derivation owed

Rests on

depends_on definition 5.37 Linear transformation ¶
- 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 4.3 Algebraic structure ¶
      - depends_on definition 4.2 Binary operation ¶
  - depends_on definition 4.32 Field ¶
    - depends_on definition 4.11 Inverse element ¶
      - depends_on definition 4.9 Neutral element ¶
    - depends_on definition 4.20 Monoid ¶
      - depends_on definition 4.9 Neutral element ¶ ↺
      - depends_on definition 4.18 Semigroup ¶
    - depends_on definition 4.30 Ring ¶
      - depends_on definition 4.8 Commutativity; abelian structure ¶ ↺
      - depends_on definition 4.28 Distributivity ¶
      - depends_on definition 4.21 Group ¶
      - depends_on definition 4.18 Semigroup ¶ ↺
  - depends_on definition 4.31 Module ¶
    - depends_on definition 4.29 Action ¶
      - depends_on definition 4.9 Neutral element ¶ ↺
    - depends_on definition 4.28 Distributivity ¶ ↺
    - depends_on definition 4.21 Group ¶ ↺
    - depends_on definition 4.30 Ring ¶ ↺

Supports

depends_on definition 5.57 Automorphism ¶
depends_on definition 5.70 Characteristic polynomial ¶
- depends_on definition 5.73 Algebraic and geometric multiplicity ¶
  - depends_on proposition 5.74 prop:lin-geometric-le-algebraic ¶
- depends_on theorem 5.71 The eigenvalues are the roots of the characteristic polynomial ¶
  - depends_on corollary 5.72 Existence of an eigenvalue over $\C$ ¶
    - depends_on theorem 5.98 Primary decomposition ¶
      - depends_on corollary 5.99 Semisimple and nilpotent parts ¶
    - depends_on theorem 5.79 Spectral theorem for a self-adjoint operator ¶
      - depends_on definition 30.20 Pressure and deviatoric stress ¶
      - depends_on proposition 30.9 Principal strains ¶
      - depends_on theorem 5.80 Simultaneous diagonalization of commuting self-adjoint operators ¶
      - depends_on theorem 5.84 Spectral theorem for a real symmetric operator ¶
      - depends_on theorem 28.33 Normal modes ¶
      - depends_on theorem 29.16 Principal axes ¶
  - depends_on proposition 32.17 Classification of planar fixed points ¶
    - depends_on example 32.18 A linear centre that is really a stable focus ¶
    - depends_on theorem 32.25 Hopf bifurcation, quoted ¶
      - depends_on proposition 32.26 The Hopf normal form ¶
  - depends_on theorem 5.84 Spectral theorem for a real symmetric operator ¶ ↺
  - depends_on theorem 9.26 Structure of the solutions ¶
    - depends_on corollary 9.27 Linear stability ¶
      - depends_on corollary 9.41 Hill's equation ¶
      - depends_on proposition 9.31 Lyapunov's first method: asymptotic stability ¶
      - depends_on proposition 9.29 The planar classification ¶
      - depends_on theorem 9.39 Floquet ¶
depends_on definition 5.76 Diagonalizable operator ¶
- depends_on corollary 5.99 Semisimple and nilpotent parts ¶ ↺
- depends_on theorem 5.77 Criterion for diagonalizability ¶
  - depends_on proposition 32.39 Stability of a periodic orbit ¶
    - depends_on phenomenon 32.87 Universal period doubling ¶
    - depends_on proposition 32.84 Fixed points and the first period doubling ¶
      - depends_on phenomenon 32.87 Universal period doubling ¶ ↺
  - depends_on theorem 32.13 Linear stability ¶
    - depends_on proposition 32.65 Fixed points of the Lorenz system and their stability ¶
    - depends_on proposition 32.23 Saddle-node bifurcation ¶
      - depends_on proposition 32.95 Intermittency: the scaling of the laminar phase ¶
    - depends_on proposition 32.24 Transcritical and pitchfork bifurcations ¶
    - depends_on proposition 28.69 Which branch is stable, and where it ends ¶
depends_on definition 5.69 Eigenvector, eigenvalue, eigenspace ¶
- depends_on corollary 9.25 Normal modes of a diagonalizable system ¶
  - depends_on corollary 9.27 Linear stability ¶ ↺
  - depends_on proposition 9.29 The planar classification ¶ ↺
- depends_on definition 5.73 Algebraic and geometric multiplicity ¶ ↺
- depends_on definition 9.30 Hyperbolic equilibrium ¶
  - depends_on theorem 9.32 Hartman–Grobman; quoted ¶
    - depends_on theorem 32.15 Hartman–Grobman, restated from Part II ¶
      - depends_on definition 32.22 Bifurcation ¶
      - depends_on example 32.18 A linear centre that is really a stable focus ¶ ↺
- depends_on proposition 5.75 Eigenvectors for distinct eigenvalues are independent ¶
  - depends_on theorem 5.77 Criterion for diagonalizability ¶ ↺
- depends_on theorem 5.77 Criterion for diagonalizability ¶ ↺
- depends_on theorem 5.71 The eigenvalues are the roots of the characteristic polynomial ¶ ↺

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`	→	Linear transformation	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:2494
`depends_on`	←	Automorphism	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:2500
`depends_on`	←	Characteristic polynomial	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3162
`depends_on`	←	Diagonalizable operator	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3361
`depends_on`	←	Eigenvector, eigenvalue, eigenspace	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3147