equation 5.74 eq:lin-matrix-rep

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Supports

• 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 proposition 5.42 The adjoint exists, is unique, and is linear ¶
  • depends_on corollary 5.82 Spectral theorem for a normal operator ¶
• depends_on proposition 5.125 prop:lin-matrix-algebra ¶
  • depends_on corollary 5.168 cor:rep-adjoint-faithful ¶
  • depends_on definition 5.162 Representation of an algebra ¶
    • depends_on definition 5.163 Faithful, equivalent, invariant ¶
      • depends_on corollary 5.168 cor:rep-adjoint-faithful ¶ ↺
      • depends_on proposition 5.164 The kernel is an ideal ¶
        • depends_on corollary 5.168 cor:rep-adjoint-faithful ¶ ↺
    • depends_on proposition 5.167 Adjoint representation ¶
      • depends_on corollary 5.168 cor:rep-adjoint-faithful ¶ ↺
    • depends_on proposition 5.164 The kernel is an ideal ¶ ↺
  • depends_on definition 9.21 Matrix exponential ¶
    • depends_on proposition 9.22 The exponential and its derivative ¶
      • depends_on theorem 9.23 Solution of a constant-coefficient system ¶
        • depends_on corollary 9.25 Normal modes of a diagonalizable system ¶
        • depends_on proposition 9.87 The distinguished limit, and which end carries the layer ¶
        • depends_on proposition 9.31 Lyapunov's first method: asymptotic stability ¶ ↺
        • depends_on proposition 9.43 The unmodulated line, and the feet of the tongues ¶
        • depends_on theorem 9.32 Hartman–Grobman; quoted ¶
        • depends_on theorem 9.26 Structure of the solutions ¶ ↺
  • depends_on proposition 5.139 Dimension of the symplectic group ¶
    • depends_on proposition 24.5 Properties of the symplectic group ¶
      • depends_on remark 24.6 Connectedness, and what it rests on ¶
      • depends_on remark 24.7 Contrast with the orthogonal group ¶
    • depends_on remark 22.21 The count ¶
• depends_on proposition 5.51 prop:lin-matrix-inverse ¶
• depends_on proposition 5.45 prop:lin-matrix-unique ¶
  • depends_on proposition 5.125 prop:lin-matrix-algebra ¶ ↺
  • depends_on proposition 5.51 prop:lin-matrix-inverse ¶ ↺
  • depends_on proposition 5.54 prop:lin-same-dim-isomorphic ¶
  • depends_on proposition 5.167 Adjoint representation ¶ ↺