definition 9.21 Matrix exponential

open in the book · parts/02-mathematical-methods/07-odes-sturm-liouville.tex:722 · p. 280

ground object -- no derivation owed

Rests on

depends_on definition 5.19 Norm ¶
- 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 ¶ ↺
depends_on proposition 5.125 prop:lin-matrix-algebra ¶
- depends_on definition 5.121 Algebra ¶
  - depends_on definition 4.4 Internal binary operation; magma ¶ ↺
  - depends_on definition 4.33 Vector space ¶ ↺
- depends_on definition 5.37 Linear transformation ¶
  - depends_on definition 4.33 Vector space ¶ ↺
- depends_on equation 5.150 eq:lin-ctesesc2 ¶
- depends_on equation 5.74 eq:lin-matrix-rep ¶
- depends_on proposition 5.45 prop:lin-matrix-unique ¶
  - 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 5.37 Linear transformation ¶ ↺
  - depends_on equation 5.29 eq:lin-indepli2 ¶
  - depends_on equation 5.74 eq:lin-matrix-rep ¶ ↺
  - proves proof ch:03-linear-algebra-representations@proof-16 ¶
- proves proof ch:03-linear-algebra-representations@proof-55 ¶

Supports

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 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 proposition 9.29 The planar classification ¶ ↺
  - depends_on proposition 9.87 The distinguished limit, and which end carries the layer ¶
    - depends_on proposition 9.88 The model problem, and the error of the composite ¶
  - 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 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 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`	→	Norm	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:738
`depends_on`	→	prop:lin-matrix-algebra	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:738
`depends_on`	←	The exponential and its derivative	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:753