theorem 5.71 The eigenvalues are the roots of the characteristic polynomial

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:3212 · p. 134

Rests on

depends_on definition 5.70 Characteristic polynomial ¶
- 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 4.32 Field ¶
      - depends_on definition 4.31 Module ¶
- 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 ¶
    - depends_on definition 5.5 Linear combination ¶
      - depends_on definition 4.33 Vector space ¶ ↺
  - 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 ¶ ↺
  - 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 5.12 Subspace generated by a set of vectors ¶
      - depends_on definition 5.5 Linear combination ¶ ↺
      - depends_on definition 5.7 Vector subspace ¶ ↺
      - proves proof ch:03-linear-algebra-representations@proof-3 ¶
    - depends_on definition 5.14 Linear independence ¶
      - depends_on definition 5.5 Linear combination ¶ ↺
  - depends_on definition 5.39 Kernel, image, nullity, rank ¶ ↺
  - depends_on lemma 5.38 Exchange and completion ¶
    - 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 ¶ ↺
    - proves proof ch:03-linear-algebra-representations@proof-11 ¶
  - 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 ¶

Supports

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 phenomenon 30.70 Hertzian contact ¶
      - depends_on phenomenon 30.60 Coulomb's law of torsion ¶
      - depends_on proposition 30.33 Conversion among the moduli ¶
      - depends_on proposition 30.51 Moment and curvature ¶
      - depends_on remark A.676 The third cubic constant is measured, and it is not small ¶
      - depends_on theorem A.695 Boussinesq's point-force solution, quoted ¶
    - depends_on definition 30.79 Maxwell viscoelasticity ¶
      - depends_on proposition 30.80 A Maxwell material is a solid or a fluid depending on how long you watch ¶
    - depends_on proposition 30.74 The maximum shear stress, and its indifference to pressure ¶
      - depends_on phenomenon 30.73 Yield and brittle fracture ¶
      - depends_on remark A.705 The dangerous stress is below the surface ¶
      - depends_on remark 30.75 Two criteria, one uncited ¶
    - 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 proposition 5.92 The eigenvalues do not control the norm ¶
    - depends_on lemma A.610 Block positivity ¶
      - depends_on proposition A.611 The constraint on $\Gamma$ ¶
    - depends_on lemma A.601 Several variables, complex symmetric matrix ¶
      - depends_on proposition A.612 The dictionary between $A$ and $\Gamma$ ¶
      - depends_on proposition A.602 The Wigner function of a Gaussian is a positive Gaussian ¶
    - depends_on lemma A.600 The real part of an inverse ¶
      - depends_on lemma A.601 Several variables, complex symmetric matrix ¶ ↺
      - depends_on proposition A.612 The dictionary between $A$ and $\Gamma$ ¶ ↺
    - depends_on proposition 5.90 Polar decomposition ¶
    - depends_on proposition 5.94 Principal axes of a real quadratic form ¶
      - depends_on proposition 5.95 The focal polar equation of a conic ¶
    - 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 corollary 5.89 Rayleigh–Ritz ¶
      - depends_on proposition 5.88 Rayleigh quotient 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 theorem 29.29 Poinsot's construction ¶
    - depends_on definition 29.35 The Lagrange top ¶
    - depends_on definition 29.18 Classification of tops ¶
      - depends_on proposition 29.28 Free rotation of a symmetric body ¶
    - depends_on phenomenon 29.31 The intermediate-axis instability ¶
      - depends_on remark 29.32 What the linearization is entitled to conclude ¶
    - depends_on proposition 29.33 Dissipation drives a free body to its largest moment ¶
    - depends_on proposition 29.17 Triangle inequalities ¶
      - depends_on proposition 29.22 Perpendicular-axis theorem ¶
    - depends_on theorem 29.25 Euler's equations ¶
      - depends_on phenomenon 29.46 A fast spinning top does not fall ¶
      - depends_on phenomenon 29.31 The intermediate-axis instability ¶ ↺
      - depends_on proposition 29.26 The two integrals of torque-free motion ¶
      - depends_on proposition 29.28 Free rotation of a symmetric body ¶ ↺
      - depends_on proposition 24.41 The free rigid body is a Lie–Poisson system ¶
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 9.43 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 ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

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Edges

type	direction	node	provenance	where
`depends_on`	→	Characteristic polynomial	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3217
`depends_on`	→	Eigenvector, eigenvalue, eigenspace	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3217
`depends_on`	→	Injective, surjective, invertible	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3217
`depends_on`	←	Existence of an eigenvalue over $\C$	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3243
`depends_on`	←	Classification of planar fixed points	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:378
`depends_on`	←	Spectral theorem for a real symmetric operator	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3697
`depends_on`	←	Structure of the solutions	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:891
`proves`	←	ch:03-linear-algebra-representations@proof-26	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3221