definition 5.69 Eigenvector, eigenvalue, eigenspace

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:3127 · p. 133

ground object -- no derivation owed

Rests on

depends_on definition 5.56 Endomorphism ¶
- 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 4.32 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 5.39 Kernel, image, nullity, rank ¶
- depends_on definition 5.37 Linear transformation ¶ ↺
- depends_on definition 5.7 Vector subspace ¶
  - depends_on definition 5.5 Linear combination ¶
    - depends_on definition 4.33 Vector space ¶ ↺
- proves proof ch:03-linear-algebra-representations@proof-12 ¶

Supports

depends_on corollary 9.25 Normal modes of a diagonalizable system ¶
- depends_on corollary 9.27 Linear stability ¶
  - depends_on corollary 9.41 Hill's equation ¶
    - depends_on proposition 9.43 The unmodulated line, and the feet of the tongues ¶
  - 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 corollary 9.41 Hill's equation ¶ ↺
- depends_on proposition 9.29 The planar classification ¶ ↺
depends_on definition 5.73 Algebraic and geometric multiplicity ¶
- depends_on proposition 5.74 prop:lin-geometric-le-algebraic ¶
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 proposition 32.23 Saddle-node bifurcation ¶
      - depends_on proposition 32.24 Transcritical and pitchfork bifurcations ¶
      - depends_on theorem 32.25 Hopf bifurcation, quoted ¶
    - 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 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 32.24 Transcritical and pitchfork bifurcations ¶ ↺
    - depends_on proposition 28.69 Which branch is stable, and where it ends ¶
depends_on theorem 5.77 Criterion for diagonalizability ¶ ↺
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 definition 30.31 The engineering moduli ¶
      - depends_on definition 30.79 Maxwell viscoelasticity ¶
      - depends_on proposition 30.74 The maximum shear stress, and its indifference to pressure ¶
      - depends_on remark 30.75 Two criteria, one uncited ¶
    - depends_on proposition 30.9 Principal strains ¶
      - depends_on example 30.10 Simple shear and pure shear are the same strain ¶
    - depends_on theorem 5.80 Simultaneous diagonalization of commuting self-adjoint operators ¶
      - depends_on corollary 5.82 Spectral theorem for a normal operator ¶
    - depends_on theorem 5.84 Spectral theorem for a real symmetric operator ¶
      - depends_on corollary 5.91 The operator norm of a real array ¶
      - depends_on lemma A.610 Block positivity ¶
      - depends_on lemma A.601 Several variables, complex symmetric matrix ¶
      - depends_on lemma A.600 The real part of an inverse ¶
      - depends_on proposition 5.90 Polar decomposition ¶
      - depends_on proposition 5.94 Principal axes of a real quadratic form ¶
      - depends_on proposition 5.92 The eigenvalues do not control the norm ¶
      - depends_on theorem 5.86 Simultaneous diagonalization of a definite pencil ¶
    - depends_on theorem 28.33 Normal modes ¶
      - depends_on example 28.39 The linear triatomic molecule ¶
      - depends_on proposition 28.40 Modes and dispersion of the loaded string ¶
      - depends_on remark 28.34 Part II owes the simultaneous diagonalization of two quadratic forms ¶
    - depends_on theorem 29.16 Principal axes ¶
      - depends_on definition 29.19 Inertia ellipsoid ¶
      - depends_on definition 29.35 The Lagrange top ¶
      - depends_on definition 29.18 Classification of tops ¶
      - depends_on phenomenon 29.31 The intermediate-axis instability ¶
      - depends_on proposition 29.33 Dissipation drives a free body to its largest moment ¶
      - depends_on proposition 29.17 Triangle inequalities ¶
      - depends_on theorem 29.25 Euler's equations ¶
- 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 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 ¶ ↺

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`	→	Endomorphism	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3147
`depends_on`	→	Kernel, image, nullity, rank	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3147
`depends_on`	←	Normal modes of a diagonalizable system	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:855
`depends_on`	←	Algebraic and geometric multiplicity	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3274
`depends_on`	←	Hyperbolic equilibrium	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1154
`depends_on`	←	Eigenvectors for distinct eigenvalues are independent	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3331
`depends_on`	←	Criterion for diagonalizability	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3370
`depends_on`	←	The eigenvalues are the roots of the characteristic polynomial	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3217